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Digitized  by  the  Internet  Archive 

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http://www.archive.org/details/courseinalgebrabOOvanvrich 


ootjk.se 


IN 


ALGEBRA 


Bking  Course  One  in  Mathematics 

IN  THE 

UniveRvSity  of 'Wisconsin. 


BY' 

C.  A.  VAN  VELZER  and  CHAS.  S.  SLIGHTER. 


MADISON,  WIS.: 

Capital  City  Pud.  Co.,  Printers, 

1888. 


Copyrighted  1888 

BY 

C.  A.  VAN  VELZER  and  CHAS.  S.  SLtCHTER. 


PREFACE 


The  present  volume  originated  in  a  desire  on  the  part  of  the 
authors  to  furnish  a  text  of  Course  I.  as  was  previously  mapped 
out  by  the  department  of  mathematics  at  the  University  of 
Wisconsin. 

The  orginal  intent  was  to  produce  a  syllabus  for  the  use  of 
students  in  this  institution,  but  it  was  subsequently  thought  that 
a  work  which  w^ould  be  useful  here  might  also  be  found  useful 
elsewhere,  and  hence  it  was  decided  to  give  the  w^ork  more  the 
character  of  a  treatise  than  a  syllabus.  To  insure  the  best  results 
it  was  thought  desirable  to  print  the  present  preliminary  edition 
and  put  it  to  the  test  of  class  room  work,  and  at  the  same  time  to 
invite  criticism  and  suggestions  from  teachers  and  others  inter- 
ested in  mathematics,  and  then  from  the  results  of  the  authors' 
tests,  and  from  the  experience  of  others,  to  rewrite  the  work, 
changing  it  freely.  For  these  reasons  the  treatment  of  many 
subjects  in  the  following  pages  should  be  understood  as  merely 
tentative.  The  final  form  will  depend  entirely  upon  the  results 
of  experience. 

An  examination  of  the  text  will  reveal  many  deviations  from 
the  beaten  path,  but  the  idea  was  not  to  deviate  simply  for  the 
sake  of  being  different  from  others  ;  on  the  contrary  the  authors 
have  freely  drawn  from  other  works.  The  sources  from  which 
material  has  been  most  largely  drawn  are  the  following:  For 
problems,  Christie's  Test  Questions  and  Wolstenholm's  Collec- 
tion ;  for  various  matters  in  the  text,  Kempt' s  Lehrbuch  in  die 
Moderne  Algebra,  and  the  algebras  of  Chrystal;  Aldis;  Hall  and 
Knight;  Oliver,  Wait  and  Jones;  and  Todhunter  ;  for  historical 
notes,  Marie's  Histoire  des  Sciences  Mathematiques  et  Physiques, 
and  Matthiesen'sGrundzuge  der  Antiken  und  Modernen  Algebra. 


mSOBOSS 


IV 

Part  II.  of  the  present  work,  containing  chapters  on  Iniagin- 
aries,  the  Rational  Integral  Function  of  .r,  Solution  of  Numerical 
Equations  of  Higher  Degree,  Graphic  Representation  of  Equa- 
tions, and  Determinants,  has  already  appeared  and  for  this  part, 
as  well  as  for  the  present  volume,  suggestions  are  invited. 

Several  modifications  have  already  suggested  themselves  to  the 
authors,  but  it  is  hoped  that  any  into  whose  hands  either  volume 
may  fall  will  communicate  with  the  authors  with  reference  to  any 
changes  before  the  work  is  put  in  permanent  form. 

University  of  Wisconsin, 
Madison,   Wis,  1888. 


TABLE  OF  CONTENTS. 


Pa  OK. 

CHAPTER  I. 
Introduction,  .  .  _  .  j 

CHAPTER  II. 
Theory  of  Indices,  -  -  -  -        i$ 

CHAPTER  III. 
Radicai.  Quantities  and  Irrational  Expressions,  35 

CHAPTER  IV. 

Quadratic     Equations     Containing     One     Unknown 
Quantity,  -  -  -  -  -        51 

CHAPTER  V. 

Theory    of     Quadratic     Equations     and     Quadratic 
Functions,  -  -  -  -  64 

CHAPTER  VI. 
Single  Equations,  -  -  -  -        74 

CHAPTER  VII. 
Systems  of  Equations,  -  -  -  89 

CHAPTER  VIII. 
Progressions,       -  -  -  -  -      ic6 

CHAPTER  IX. 

Arrangements  and  Groups,      -  -  -  114 

CHAPTER  X. 
Binomial  Theorem,  -  -  -  -      130 


VI 

CHAPTER  XI. 
Theory  of  Limits,       -  -  -  -  139 

CHAPTER  Xn. 
Undetermined  Coefficients,  -  -  -      153 

CHAPTER  XIII. 
Derivatives,  -  -  -  -    .  164 

CHAPTER  XIV. 
Series,   -  -  -  -  -  -      183 

CHAPTER  XV. 
Logarithms,  -  -  -  -  196 


ALGEBRA 


CHAPTER    I. 

INTRODUCTION. 

1,  Definitions.  When  we  wish  to  use  a  general  term  which 
shall  inclnde  in  its  meaning  any  intelligible  combination  of  alge- 
braic symbols  and  quantities,  the  word  Expression  will  be  adopted. 
Thus 

,2      j\  r     2  I  i     I    \       c^ -{■  d^ -\- abed 

(x^—d)  Cax^-\-bx-\-e);  — - — — ^ \/h^,-l.   / — 

may  be  called  expressions.  It  includes  the  words  polynomial, 
fraetio7i,  and  radical  and  more  besides. 

When  we  wish  to  call  attention  to  the  fact  that  certain  specified 
quantities  appear  in  an  expression  it  may  be  called  a  Function  of 
those  quantities.  Thus  if  we  desire  to  point  out  that  x  appears  in 
the  first  expression  above,  it  would  be  called  a  ftinction  of  x.  If 
we  wish  to  state  that  a,  b,  c,  and  d  occur  in  the  second  expression, 
we  would  call  it  2,  fimction  of  a,  b,  c,  and  d.  If  we  wish  to  say 
that  y  occurs  in  the  last  expression,  it  may  be  called  o.  function  of 
y,  or  if  we  wish  to  say  that  a,  b,  and  y  occur  in  it,  we  would 
speak  of  it  as  afmction  of  a,  b,  and  y.  A  formal  definition  of  the 
word  function  would  be  : 

A  Function  of  a  quantity  is  a  name  applied  to  any  mathemati- 
cal expression  in  which  the  quantity  appears. 

2.  Definition.  An  expression  is  Integral  with  respect  to  any 
quantity  or  quantities,  that  is,  is  an  integral  function  of  those 
quantities,  when  the  quantities  named  do  not  appear  in  any  man- 
ner as  divisors.  Thus  z^x^-\-\x—y/2  is  integral  with  respect  to 
X ;  that  is,  is  an  integral  function  of  x. 

a—b-\-a-b     ab 

x^-\-xy        X 

is  integral  with  respect  to  a  and  b,  but  fractional  with  respect  to 


2  Algebra. 

jc  and  J/ ;  that  is,  is  an  integral  function  of  a  and  d,  but  a./radwfial 
function  of  jr  and  y,  the  word  fractional  meaning  just  the  opposite 
to  integral. 

3.  Definition.  An  Expression  is  Rational  with  respect  to 
any  quantity  or  quantities,  or  is  a  rational  function  of  those  quan- 
tities, when  the  quantities  referred  to  are  not  involved  in  any 
manner  by  the  extraction  of  a  root.     Thus 

i^TlT^ 

is  rational  with  respect  to  x,  but  irrational  with  respect  to  c  and 
d\  that  is,  it  is  a  rational  function  of  x,  but  an  irrational  function 
of  c  and  d,  the  term  irrational  being  used  in  just  the  opposite 
sense  from  rational. 

4.  An  expression  may  be  both  rational  and  integral  with  re- 
spect to  certain  quantities,  in  which  case  it  may  be  spoken  of  as  a 
Rational  Integral  Expression  with  respect  to  those  quantities,  or 
as  a  rational  integral  function  of  the  quantities.  In  the  sam6 
way  we  may  speak  of  an  expression  which  is  rational  and  frac- 
tional with  respect  to  certain  quantities  as  a  Rational  Fractional 
Expression  with  reference  to  those  quantities,  or  as  a  rational 
fractional  function  of  the  quantities.  In  like  manner  we  may  use 
the  terms  Irrational  Integral  Expression  and  Irrational  Fractional 
Expression,  or  Irrational  Integral  Function  or  Irrational  Frac- 
tional Function. 

In  the  following  examples  the  student  is  expected  to  answer 
the  question.  What  kind  of  an  expression?  with  reference  to  the 
quantities  specified  opposite  each. 

/.     ax^-\-a^x''-\-a^x.  With  respect  to  ;t- ?  to  a ?  to  Jt:  and  a~i 


c 

2. 

2« 


(  \-\-—y  I  ~ ^)  With  respect  to  <2 ?  to  r? 


a 


J.     bx" T+^y-  With  respect  to  jt  ?  to  j^' ?  to  ;t:  and  j'? 

>/a  x^-+^b+\/c       With  respect  to  ;»;?  to^y? 

ay^-\-by-\-c       '  to  x  andjj'?  to  a,  b,  and  r? 

5.  Definition.     If  by  any  operation  we  render   an   expres- 
sion integral  with  reference  to  certain  quantities,  in  respect  to 


Introduction.  5 

which  it  was  previously  fractional,  we  are  said  to  Integralize  the 
expression  with  respect  to  those  quantities.     Thus  the  expression 

a'x'     2ab      b'x' 
b         x^         a 
is  integralized  with  respect  to  a  and  b  if  it  is  multiplied  by  ab. 

Similarly,  if,  by  any  operation,  we  .  render  an  expression 
rational  with  reference  to  certain  quantities,  in  respect  to  which 
it  was  previously  irrational,  we  are  said  to  Rationalize  the  expres- 
sion with  respect  to  the  quantities  named.  Thus  if  we  square 
the  irrational  expression 

'^^ x'-\r^~ab  xy-^-y'' 
it  is  rationalized  with  respect  to  x  and  y. 

6.  Definitions.  The  Degree  of  a  term  with  respect  to  any 
quantity  or  quantities  is  the  sum  of  the  exponents  of  the  quan- 
tities named.  Thus  ab'^x^y  is  of  the  third  degree  with  reference  to 
X,  of  the  first  degree  with  reference  to  y,  of  the  fourth  degree  with 
reference  to  x  and  y,  of  the  third  degree  with  reference  to  a  and 
b,  etc.  But  the  degree  with  reference  to  any  quantities  is  not 
spoken  of  unless  the  term  is  rational  and  integral  with  respect  to 
those  quantities.     Thus  we  do  not  speak  of  the  degree  of  such  a 

term  ^^      — ,  with  respect  to  either  a  or  x. 

The  Degree  of  a  polynomial  with  respect  to  any  specified  quan- 
tities is  the  degree  of  that  one  of  its  teniis  whose  degree  (with  re- 
spect to  the  same  quantities)  is  highest.  Thus,  x^—alxy^-^-cxy 
is  of  the  third  degree  with  respect  to  x,  of  the  second  degree  with 
respect  to  y,  and  of  the  fourth  degree  with  respect  to  x  and  y. 
But  the  degree  of  a  polynomial  is  not  spoken  of  unless  the  poly> 
nomial  is  rational  and  integral  with  respect  to  the  quantities 
specified. 

It  can  easily  be  seen  that  the  degree  of  the  product  of  several 
polynomials  is  the  sum  of  their  separate  degrees.     Thus 

{^x'-^xy-\-f)  (xy-j-bx^y) 
is  of  the  fifth  degree  with  respect  to  x  and  y ;  of  what  degree  is  it 
with  respect  to  jf  ?  with  respect  to^? 

The  Degree  of  an  Equation  is  the  degree  of  the  term  of  highest 
degree  with  respect  to  the  tinkyiown  quantities.     But  both  mem- 


4  Algebra. 

bers  of  the  equation  must  be  rational  and  integral  with  respect  to 
the  unknown  quantities  and  the  indicated  operations  must  be  per- 
formed ;  otherwise  the  degree  is  not  spoken  of. 
What  is  the  degree  of 

(ji"  +y )  (xy  -}- 1 ) = 2o8jf  y  ? 

Instead  of  speaking  of  expressions  as  being  of  the  first  or  of  the 
second  or  of  the  third  degree,  they  are  commonly  designated  by 
adjectives  borrowed  from  geometry  as  linear  or  quadratic  or  cubic 
expressions  respectively.  An  expression  of  the  fourth  degree  is 
sometimes  called  bi-quadratic,  meaning  twice  squared. 

In  place  of  the  expression,  "of  the  second  degree  in  respect  to 
.r,"  it  is  common  to  say,  "of  the  second  degree  in  jr." 

7.  DefinitioNvS.  a  polynomial  is  Homogeyieous  with  respect 
to  certain  quantities  when  all  its  terms  are  of  the  same  degree 
with  respect  to  those  quantities.  Thus  a^-\-a^b-\-ab'-\-b^  is  homo- 
geneous with  respect  to  a  and  b. 

An  equation  is  Homogeneous  when  all  the  terms  are  of  the  same 
degree  with  reference  to  the  unknown  quantities.  Thus  the  equa- 
tion xy-\-j>^-\-x^=o  is  homogeneous,  but  xy-^y^-\-x^=2o  is  not 
homogeneous. 

It  is  to  be  noted ^  here  that  we  use  the  term  homogeneous  equa- 
tion in  the  strict  sense,  following  the  established  use  of  the  term. 
But  in  some  American  text  books  homogeneous  equation  includes 
equations  like  x-y-\-y^-{-x^=20,  that  is,  no  account  is  taken  of 
terms  involving  nothing  but  known  quantities. 

8.  Definition.  An  expression  is  Symmetrical  with  respect 
to  two  quantities  if  the  expression  is  unaltered  when  the  two 
quantities  are  interchanged.  Thus  x^-{-y^  is  symmetrical  with  re- 
spect to  X  and  J' ;  for  putting  j/  for  x  and  x  for  j  we  obtain  j^+j*:^, 
which  is  the  same  as  the  original.  Also  x" -\- ax + a''  is  symmet- 
rical with  respect  to  a  and  x.  Is  x'^  +  2xv—y  symmetrical  with 
reference  to  x  and  y  ? 

An  equation  of  two  unknown  quantities  is  symmetrical  when 
the  interchange  of  the  unknown  quantities  throughout  does  not 
modify  the  equation.     Such  is 

X + xy  -f  xy'  +y=  1 024. 


Introduction.  5 

9.  Incommensurable  Numbers.  Algebraic  numbers*  may 
be  divided  into  two  kinds,  depending  upon  the  relation  which 
they  bear  to  the  unit  or  unity.  If  a  number  has  a  common 
measure  with  unity,  it  is  called  a  commensurable  number.  Thus 
7  is  a  commensurable  number ;  also  J  is  a  commensurable  number, 
since  one  quarter  of  the  unit  is  a  common  measure  between  f  and 
unity.  Commensurable  numbers  thus  include  both  integers  and 
fractions.  If  a  number  has  no  common  measure  with  unity,  it  is 
called  an  incommensurable  number.  Thus  >/~2  is  incommen- 
surable. A  little  consideration  will  show  that  v^  2  cannot  be  an 
integer  nor  a  fraction.  It  is  not  an  integer  because  (0)^=0, 
(1)^=1,  and  (2)^=4,  and  there  are  no  integers  intermediate  be- 
tween these.     It  cannot  be  a  fraction,  for  if  possible  suppose  that 

some  irreducible  fraction,  represented  by—,  equals  v^  2  .     Then 

"^  2  =  -7-,  or  squaring,  2——^,  which  is  absurd,  for  an  integer 

cannot  equal  an  irreducible  fraction.  Therefore  -v^  2  is  not  a 
fraction.  But  it  is  an  exact  quantity,  for  we  can  draw  a  geomet- 
rical representation  of  it.  Take  each  of  the  two  sides,  CA  and 
CB,  of  a  right  angled  triangle  equal  to  i.  Then  AB,  the  hypoth- 
enuse,willequalN^(i)^  +  (i)^=N^  2  Thus  A^ 
v^  2  is  the  exact  distance  from  A  to  B, 
which  is  a  perfectly  definite  quantity. 
Thus  the  idea  that  incommensurables  are 
indefinite  or  inexact  must  be  avoided,  (l) 
This  notion  has  arisen  because  the  frac- 
tions we  often  use  in  place  of  incommen- 
surables, such  as  1. 4 142-}-  for  v/  2,  are 
7nerely  approximations  to  the  true  value.    (^  ("fj  ^ 

We  now  give  a  property  of  incommensurable  numbers  which 
will  serve  to  make  their  separation  from  the  class  of  commensur- 
able numbers  (integers  and  fractions)  more  apparent.  It  is  that 
an  incommensurable  number  when  expressed  i?i  the  decimal  scale 
never  repeats,  while  a  comtnensurable  number  so  expressed  always 
repeats. 

*  As  here  used  the  term  Algebraic  number  does  not  include  the  so-called  imaginaries, 
which,  strictly  speaking,  are  not  numbers  at  all.  Imaginaries  ai-e  treated  in  Chapter  1. 
Part  II. 


6  Algkbra^  ^^        I  (  ^A«     ; 

Thus,  75=75.0000000000+  repeating  the  o. 

^=     .5000000000+  repeating  ike  o. 
i=     -3333333333+  repeating  the  3. 
yyy=     .279279279279+  repeating  the  279. 
>/  3  =   2.7320508+  never  repeating , 
"^20=   2. 7 1 441 77+  7iever  repeating. 
-•=.   3.1415926+  Wd'Z'^r  repeating. 
The  student  should  endeavor  to  get  a  fair  notion  of  what  is 
meant  by  an  incommensurable  number.     It  is  a  difficult  idea  to 
grasp  at  once,  but  it  is  one  which  the  student  should  continue  to 
consider  until  the  conception  takes  a  definite  and  rational  shape. 

POSITIVE   AND   NEGATIVE   QUANTITIES. 

10.  In  Algebra  we  are  often  called  upon  to  distinguish  between 
quantities  which  are  directly  opposite  each  other ;  as,  for  instance, 
degrees  above  zero  from  degrees  below  zero  on  a  thermometer  scale, 
distance  north  of  the  Equator  from  distance  south  of  the  Equator, 
distance  east  of  a  given  point  from  distance  west  of  the  same  given 
point,  etc. 

The  distinction  is  made  by  means  of  the  signs  +  and  — ,  e.  g.^ 
if  +10°  means  a  temperature  of  10°  above  zero,  then  —10°  would 
mean  a  temperature  of  10°  below  zero,  and  if  + 10  miles  means  10 
miles  w^r///  of  the  Equator,  then  — 10  miles  would  mean  10  miles 
south  of  the  Equator,  and  if  +10  rods  means  10  rods  east  of  a 
given  point,  then  — 10  rods  would  mean  10  rods  west  of  the  same 
given  point,  and  if  +10  be  ten  units  of  any  kind  in  a^iy  sense, 
then  —10  would  be  ten  units  of  the  same  kind  in  just  the  opposite 
sense. 

These  two  kinds  of  quantities  are  called />^«VzV^  and  negative. 

11.  The  distinction  between  positive  and  negative  quantities  is 
made  by  means  of  the  same  signs  as  are  used  to  denote  the  opera- 
tions of  addition  and  subtraction,  and  it  might  seem  that  it  is  un- 
fortunate and  unnatural  that  the  same  signs  are  used  in  these  two 
ways.  It  may  be  unfortunate,  but  it  is  not  unnatural,  as  we  pro- 
ceed to  show. 

12.  Suppose  that,  by  one  transaction,  a  man  gained  $500,  and 
by  another  he  lost  $700 ;  then  he  lost  all  he  gained  and  $200  more, 


Positive  and  Negative  Quantities.  7 

or  his  capital  suffered  a  diminution  of  $200.  If  his  original 
capital  was  $1,000,  then  the  first  transaction  increased  it  to 
$1,500,  and  the  second  transaction  diminished  it  to  $800.  Thus 
an  addition  of  $500  followed  by  a  diminution  of  $700  is  equivalent 
to  a  single  diminution  of  $200,  or 

$i,ooo-}-$50o— $700= $1,000— $200. 

Hence  %^qo—%'j 00  when  joined  to  $1,000  may  be  replaced  by 
—  $200  joined  to  $1,000. 

Now,  as  any  other  original  capital  would  have  answered  as  well 
as  $1,000,  we  may  neglect  that  original  capital  and  write 
$500— $700=  —$200. 

Thus  we  see,  by  this  illustration,  that  it  is  natural  to  prefix  the 
minus  sign  to  the  $200  to  indicate  a  resultant  loss  of  $200. 

13.  We  might  have  used  an  illustration  involving  some  other 
kind  of  quantity  than  money,  as  titne,  distance,  etc.,  and  have  ob- 
tained an  equation  similar  to  the  one  just  written.  We  may  then 
make  an  abstraction  of  the  $  sign  and  write  simply 

500—  700=  —  200. 

14.  In  Ari  theme  tic  we  are  concerned  only  with  the  quantities 

o,  I,  2,  3,  4,  etc., 
and  intermediate  quantities,  but  in  Algebra  we  consider  besides 
these  the  quantities 

o,  —I,  —2,  —3,  —4,  etc., 
and  intermediate  quantities. 

15.  We  may  represent  these  two  classes  of  quantities  on  the 
following  scale, 

—5.-4.  —3.  —2,  —I,  o,  I,  2,  3,  4,  5, 

which  extends  indefinitely  in  both  directions  from  zero. 

The  sign  -f  perhaps  ought  to  precede  each  of  the  quantities  at 
the  right  of  o  in  this  scale,  but  when  no  sign  is  written  before  a 
quantity  the  +  sign  is  always  understood. 

16.  Quantities  to  the  right  of  o  in  the  above  scale  are  positive 
and  those  to  the  left  of  o  are  negative,  or  we  might  say  Arabic 
nuynerals  preceded  by  a  +  sign  or  by  no  sign  at  all  are  positive 
quantities,  and  Arabic  numerals  preceded  by  a  —  sign  are  yiegative 
quantities. 


8  Algebra. 

17.  In  Algebra  quantities  are  represented  by  letters,  but  a  letter 
is  just  as  apt  to  represent  a  quantity  to  the  left  of  o  in  the  above 
scale  as  it  is  to  represent  one  to  the  right  of  o ;  so  that,  while  in 
the  case  of  a  numerical  quantity,  /.  e.  one  represented  by  figures, 
we  can  tell  whether  the  quantity  represented  is  positive  or  nega- 
tive by  the  sign  preceding  it,  yet,  it  the  case  of  a  literal  quantity, 
/.  e.  one  represented  by  letters,  we  cannot  tell  by  the  sign  before 
it  whether  the  quantity  represented  is  positive  or  negative. 

If  we  speak  of  the  quantity  5  we  know  that  it  is  positive,  but  if 
we  speak  of  the  quantity  a  we  do  7iot  know  by  the  sign  before  it 
whether  it  is  positive  or  negative. 

We  know  that  —5  is  negative,  but  we  do  7iot  know  that  —a  is 
negative. 

A  mifuis  sign  before  a  letter  always  represents  a  quajitity  of  the 
opposite  kind  from  that  represented  by  the  same  quantity  with  a  plus 
sign  or  no  sig7i  at  all  before  it.  Thus,  if  «=3,  then  — <2=— 3,  and 
if  ^=  —  3,  then  — a  =  3. 

18.  Looking  at  the  above  scale  it  is  evident  that  of  any  two 
positive  quantities  the  one  at  the  right  is  greater  than  the  other  or 
the  one  at  the  left  is  less  than  the  other,  e.  g.  io>6  or  6<io. 

Now  it  is  found  convenient  to  extend  the  meaning  of  the  words 
* '  less  than ' '  and  ' '  greater  than ' '  so  that  this  same  thing  shall  be 
true  throughout  the  whole  scale. 

Thus  we  would  say  that 

—  5<  — 3  and— 2<o. 
It  should  be  carefully  noticed  that  this  is  a  technical  use  of  the 
words  '  *  greater  than ' '  and  '  *  less  than ' '  and  conforms  to  the  pop- 
ular use  of  these  words  only  when  the  quantities  are  positive. 

Of  course  it  would  be  wrong  to  say  that  —2  is  less  than  o  if  we 
use  "less  than"  in  the  popular  sense,  because  no  quantity  can  be 
less  than  nothing  at  all,  in  the  popular  sense  of  "  less  than." 

In  objecting  to  the  use  of  the  words  "  less  than"  in  the  popular 
sense.  Prof.  De  Morgan,  one  of  the  great  mathematicians  of  Eng- 
land, says  :  "  The  student  should  reject  the  definition  still  some- 
times given  of  a  negative  quantity  that  it  is  less  than  nothing.  It 
is  astonishing  that  the  human  intellect  should  ever  have  tolerated 
such  an  absurdity  as  the  idea  of  a  quantity  less  than  nothing ; 


Introduction.  9 

above  all,  that  the  notion  should  have  outlived  the  belief  in 
judicial  astrology,  and  the  existence  of  witches,  either  of  which 
is  ten  thousand  times  more  possible." 

This  strong  language  is  directed  against  the  use  of  the  words 
*'less  than"  in  the  popular  sense,  but  let  the  student  keep  in 
mind  that  the  words  are  used  in  a  technical  sense  and  there  will  be 
no  objection  to  such  an  inequality  as  —  2<o. 

Illustrations. —  If  we  speak  of  temperature  as  indicated  by  a 
thermometer  scale,  then  ''greater  than''  means  higher  and  ''less 
than ' '  means  lower.  If  we  speak  of  distance  east  and  west  and 
agree  that  distances  measured  east  are  positive,  then  "greater 
than  ' '  means  ' '  east  of ' ,  and  ' '  less  tha7i ' '  means  ' '  zvest  of'\  If  we 
agree  that  distances  measured  north  are  positive  and  those  meas- 
ured south  are  negative,  then  "  greater  than''  means  "north  of" , 
and  ' '  less  tha?i ' '  means  ' '  south  of" ,  etc. 

THE   RULE   OF   SIGNS    IN    MULTIPLICATION   AND    DIVISION    IN 

ALGEBRA. 

19.  If  we  take  a  and  b  any  two  positive  quantities,  it  is  easy  to 
see  that  the  notion  of  multiplication  we  get  from  arithmetic  will 
enable  us  to  deal  with  any  case  of  multiplication  where  the  w///- 
tiplier  is  a  positive  quantity,  for,  evidently,  a  can  be  repeated  b 
times,  and  so  can  —a  be  repeated  b  times,  but  a  cayinot  be  repeated 
—  b  times,  e.  g.  3,  and  also  —3,  can  be  repeated  5  times,  but  3 
cannot  be  repeated  —5  times. 

Thus,  when  the  mnltiplier  is  negative,  multiplication  has  no  7nean- 
ing  according  to  the  arithmetical  notion  of  multiplication,  and  so  we 
are  obliged  to  broaden  our  ideas  of  multiplication  in  some  way  or 
else  exclude  the  operation  when  the  multiplier  is  negative. 

20.  The  primar>'  definition  of  multiplication  is  repeated  addi- 
tion, yet,  even  in  arithmetic,  the  word  outgrows  its  original 
meaning,  for,  by  no  stretch  of  language,  can  the  operation  of  mul- 
tiplying I  by  ^  be  brought  under  the  original  definition. 

According  to  the  original  definition,  multiplication,  in  arithme- 
tic, is  intelligible  so  long  as  the  multiplier  is  a  whole  number. 

3  can  be  repeated  4  times,  and  so  can  \  be  repeated  4  times 
but  4  cayinot  be  repeated  \  a  time. 

A— 2- 


lo  Algebra. 

^  repeated  4  times  is  ^  multiplied  by  4,  yet,  in  arithmetic,  4 
multiplied  by  ^  is  a  familiar  operation. 

Let  us  inquire  how  this  comes  to  have  a  meaning,  and  how  it 
happens  that  4  multiplied  by  \  turns  out  to  be  ^  <^4. 

21.  As  long  as  a  and  b  are  positive  whole  numbers  it  is  easy  to 
see  that  a b=ba. 

Suppose,  to  fix  the  ideas,  that  <3!=3  and  b=^,  then  we  may 
write  down  5  rows  of  dots  with  three  dots  in  each  row,  thus — 


and  we  have  in  all  5  times  3  dots.  But  we  may  look  at  vertical 
rows  instead  of  horizontal  ones  and  we  see  three  rows  with  5  dots 
in  each  row,  and  of  course  the  number  of  dots  is  the  same  ;  so  we 
may  say 

5x3=3x5. 
Any  other  positive  whole  numbers  would  do  as  well  as  3  and  5, 
and  so  if  a  and  b  are  any  positive  whole  numbers, 

ab=ba, 
i.  e. ,  in  the  product  of  two  numbers,  if  is  indiffereyit  which  is  the 
multiplier  and  which  the  multiplicand,  so  long  as  both  numbers  are 
integers.  ^ 

2.2..  Now,  in  arithmetic,  the  operation  of  multiplication  is  so 
extended  that  eve^i  when  one  of  the  quayitities  is  a  fraction  it  shall 
still  be  indifferent  which  of  the  two  quantities  is  the  multiplier  and 
which  the  multiplicand. 

This  gives  a  7nea7iing  to  multiplication  when  the  multiplier  is  a 
fraction,  and  thus  it  happens  that  4  multiplied  by  \  is  take?i  to 
mean  the  same  as  \  multiplied  by  4. 

23.  In  exactly  the  same  way  in  algebra,  the  operation  of  mul- 
tiplication is  extended  so  that  whatever  numbers,  positive  or  neg- 
ative, integral  or  fractional,  are  represented  by  a  and  b  we  shall 
always  have 

ab'^ba. 


Introduction.  i  i 

and  since  we  know  what  is  meant  by  —3  multiplied  by  5,  the 
equation  ab=^ba  gives  a  meaning  to  5  multiplied  by  —3. 
.•.5  multiplied  by  —3=  — 15. 
From  this  we  are  led  to  say  that  when  the  multiplier  is  neg- 
ative, the  product  is  just  the  opposite  from  what  it  would  be  if 
the  multiplier  were  positive. 

Therefore,  if  a  and  b  are  any  two  positive  quantities,  we  may 
WTite  the  following  four  equations  : 

a.b=ab  (i) 

{—a).b=—ab  (2) 

a.{—b)=-—ab  (3) 

{-a).{-b)  =  ab.  (4) 

From  the  ist  and  4th  we  conclude  that  the  product  of  two  posi- 
tive quantities  or  tzvo  7iegative  qua?itities  is  positive,  and  from  the  2d 
and  3d,  the  product  of  07ie  positive  and  o?ie  negative  quantity  is  neg- 
ative. 

24.  The  four  equations  just  written  are  true  whether  a  and 
b  are  positive  nor  not. 

Consider,  for  example,  the  second  equation  under  the  supposi- 
tion that  a  is  negative  and  b  positive  ;  then  {—a).b  becomes  the 
product  of  two  positive  quantities  and  is  therefore  positive,  but 
—ab  is  also  positive  in  this  case,  as  it  should  be,  rendering  the 
equation  still  true.  And  so  of  the  other  equations,  whether  a  and 
b  are  positive  or  not.  Therefore,  directing  our  attention  to  the 
signs,  we  may  say  that  the  product  of  two  quantities  preceded  b}- 
like  signs  is  a  quantity  preceded  by  the  +  sign,  and  the  product 
of  two  quantities  preceded  by  unlike  signs  is  a  quantity  preceded 
by  a  —  sign. 

This  statement  is  usually  shortened  into  the  following — 

In  multiplication  like  sig?is  give  plus  and  imlike  sig-fis  give 
minus. 

This  is  often  confused  with  the  statement  in  italics  in  the  pre- 
ceding article.     They  are  not  identical,  but  both  are  true. 

25.  As  division  is  the  inverse  of  multiplication,  it  easily  fol- 
lows that  the  quotient  of  tivo  positive  or  two  negative  quantities  is 
positive,  and  that  the  quotient  of  a  positive  by  a  negative  quantity, 


12  AI.GEBRA. 

or  a  negative  by  a  positive  quantity,  is  yiegative.     It  also  follows 
that  in  division  like  signs  give  plus  and  unlike  signs  give  mi7ius. 
The  proof  of  these  two  statements  is  left  as  an  exercise  for  the 
student. 

26.  Theorem.    The  difference  between  like  powers  of  two  quanti- 
ties is  exactly  divisible  by  the  difference  of  the  quantities  themselves. 
It  is  easily  seen  on  trial  that 

ia' — jr')-i-(a — x)  =  a-^x. 
{a^ — x^)-7-(a—x)==a^-\-ax-hx'. 
{a'—x')^{a—x)=^d'-\-a'x-^ax^  +  x\ 
(a^—x^)-^(a—x)=a*-\-a^x-\-a^x^-\-ax^-\-x*. 
Observing  the  uniform  law  in  these  results  it  would  be  at  once 
suggested  that  the  theorem  is  universally  true ;  that  is,  that  what- 
ever be  the  value  of  n, 

^~J^.^a"-'-\-a"-^x-{-a"-'x'-\-  .    .  -^-a^x^'-'+ax^-'-^-x"-'.        (i) 
a—x 

This  can  easily  be  shown  to  be  true,  for  multiplying  the  right 

hand  side  of  this  equation  by  a—x  it  becomes  a"—x",  as  follows  : 

a"-'-}-a"-'x-^a"-'x'-{-  .    .    .  -j-a'^x"-^-^  ax"-'+x"-' 

a—x 

a'' 


'"    -\-a" 
—a" 

-'x-j-a"-'x'-h  . 
~'x—a"~''x^—  .    , 

.    .  -\-a'x''-'-\-a'x"-'-\-ax"- 
.    .  —a'x"-^—a'x"-'—ax"- 

I 

'  —  X' 

:«    -ho 

+o         +  .    . 

. .  .  -f-o         -f-o         +0 

— X'' 

But  multiplying  the  left  hand  side  of  equation  (i)  by  a — x  we 
obtain  a"—x"  also.     Hence  equation  (i)  reduces  to 

a" — x"=a" — x'\ 
and  hence  must  be  correct. 

27.  Theorem.      The  difference  of  like  even  powers  is  exactly 
divisible  by  the  sum  of  the  quantities  themselves. 
It  will  be  found  on  actual  division  that 

{a" — x^^  —  {a-\-  x)=:=a— X . 

(a*—x^)-7-(a-\-x')=a^—a-x-j-ax^—x^. 

(a'—x^)-i-(a-\-x)=a'—a*x-\-a'x'—a'x'-\-ax*—x\ 


Introduction.  13 

The  obvious  uniformity  in  these  results  forces  the  suggestion 
that  the  law  of  fonnation  of  the  quotient  will  hold  in  any  similar 
case.    That  is,  that 

^  ~'^  =a"-'—a"-'x-\-a"-^x'—  .    .    .  —a'x"-'-^ax"-''—x"-\        (i) 
a-\-x 

where  we  have  given  the  —  sign  to  the  odd  powers  of  x,  n  being 
any  even  number.  Multiplying  the  right  hand  side  of  the  equa- 
tion by  a-h^  we  obtain  a"—x",  thus: 

a"-'—a"-'x-j-a"-'x'—  .    .    .  —a'x"-'-\-ax"-'—x"-' 

a-\-x 
a"    —a"~'x-\-a"~''x^^  .    .    .  —a^x"~^-\-a''x"'~^—ax"~' 

-{■a"-'x—a"-'x'+  .    .    .  -\-d'x"-^—a'x"-'-\-ax"-'—x" 
a"    +0        +0         -f   .    .    .  +0  +0  -fo        —x" 

But  multiplying  the  left  hand  side  of  equation  (i)  by  a-f  .r  we 
obtain  a"— x"  also.  Hence  equation  (i)  must  be  true,  since  it 
reduces  to 

a"—x"=a"—x". 


28.  Theorem.  The  sum  0/ like  odd  powers  of  two  quantities  is 
exactly  divisible  by  the  sian  of  the  quayitities  themselves. 

By  trial  we  find  this  theorem  holds  in  the  first  few  cases  as 
follows : 

{a-\-x)-^(a-^x)=\. 
{a^-{-  x^)  -r-(a-^  x)  =  a'—ax  -\-  x'' . 
(a'-j-x')^(a-^x)=a*—a'x-\-a'x'—ax^+.t'. 
(a'+x')--r(a+x)=a^—a^x-^a'x'—a'x'-\-a'x-'—ax^-\-x^. 

The  simple  law  in  the  formation  of  these  results  would  naturally 
suggest  the  general  truth  of  the  theorem.     That  is,  that 

^^"=a"-'-a"-'x-\-a"-'x-'-  .    .    .  -\-a'x"-'-ax'-'-^x'-\       (i) 
a-f-x 

where  the  terms  containing  the  odd  powers  of  x  have  the  minus 
sign,  n  being  any  odd  number.    Multiplying  this  equation  through 

by  a-{-x  it  becomes 

a"-hx"=a"-\-x", 
and  hence  must  be  true. 


14  Algkbra. 

29.  The  last  three  theorems  have  such  a  variety  of  applications 

that  it  is  important  that  they  should  be  committed  to  memory. 

We  suggest  the  following  scheme  for  keeping  them  in  mind : 

x—a  divides  the  difference  of  like  powers. 

,.   .'       ^,      (  difF(fr<?nc6' of  like  ^^^n  powers, 
jf-f-^  divides  the -.  _,.,        ,,  ^ 

(  sum  oi  like  odd  powers. 

The  two  cases  which  x-{-a  divides  can  be  kept  distinct  from 
one  another  by  noticing  that  the  words  differ ence  and  even,  which 
go  together,  are  the  words  which  contain  the  ^'s. 


CHAPTER   II. 

THEORY   OF   INDICES. 

1.  By  the  definition  of  a  power  of  a  number,  «"  equals  the  con- 
tinued product  of  n  factors  each  equal  to  a. 

a"=a  a  a  a to  w  factors, 

71  being  a  positive  whole  number. 

2.  To  find  the  product  of  two  powers  of  the  same  letter. 

a^=.a  a  a, 

.'.  a?a'=^a  a  a  a  a=a^. 
Again,  a^=^a  a  a  a  a, 


.'.  a^a^=a  a  a  a  a  a  a  a=^ar. 
In  each  case  it  is  to  be  noticed  that  the  exponent  of  the  pro- 
duct equals  the  sum  of  the  exponents  of  the  two  factors. 
In  general,  if  n  and  r  are  a?iy  positive  whole  numbers, 

a"=a  a  a  a to  n  factors, 

a''=^a  a  a  .    ■,    .    .    .    .  to  r  factors. 

.',  a"a''=^a  a  a  a to  (n-\-r)  factors=«"^^ 

In  the  present  chapter  the  formula, 

a"a'-=-a"+'',  (a) 

will  be  referred  to  as  formula  a. 

This  may  be  expressed  in  words  thus — 

The  product  of  tivo  powers  of  a  quantitiy  is  equal  to  that  quantity 
with  an  exponent  equal  to  the  suui  of  the  exponents  of  the  two  fac- 
tors. 

3.  We  may  also  find  the  product  of  the  same  powers  oi  di^^r^nt 
quantities. 

a'b''=a  a  b  b^(al)(ab)=(abT\ 
alvSO  a^b^=-a  a  a  b  b  b=(ab)(ab)(ab)=(ab)^. 

And  so  in  general, 

a"b"=a  a  a  .    .    .  to  n  factors  X  ^  b  b  .    .    .  to  w  factors, 

=  (ab)(ab)(ab)  .    .    .  to  n  factors,  each  of  which  is  ab, 

=  (ab)". 

.-.  a"b"=(ab)". 


1 6  Algebra. 

4.  In  the  case  just  considered  we  have  the  sa^ne  power,  but  a 
and  b  may  be  diiferent.  In  Art.  2  we  had  the  same  letter,  but  the 
powers  may  have  been  different.  The  student  should  net  con- 
fuse these  two  cases. 

5.  The  equation  a"a''—a"^''  may  be  extended.  Multiplying 
both  sides  of  the  equation  by  a^  we  have 

.*.  a"a''a^=a"^''a^. 
By  equation  («)  the  right  hand  member  equals  a"+'+^. 

a"a''a^—a"-^''^'\ 
and  so  on,  evidently,  for  any  number  of  factors. 
Now  the  exponents  n,  r,  /  may  all  be  the  same. 
.-.  a"a"=a''\  or  (a"f^ar'\ 
and  a"a"a"=d^'\  or  (a" )^=a}", 
and  so  on.     Therefore,  evidently, 

(a"y=a"^\  {b) 

n  and  r  being  any  positive  whole  numbers.     This  formula  will  be 
referred  to  as  formula  {b). 

6.  The  equation  a"b"=(ab)"  may  be  extended.  Multiplying 
both  sides  of  this  equation  by  c"  we  obtain 

a"b"c"=(ab)"c"=(abc)", 
and  so  on,   evidently,   for  any  number  of  factors.     Hence,   the 
product  of  the  nih.  powers  of  any  number  of  quantities  is  equal  to 
the  nih.  power  of  the  product  of  those  quantities. 

EXAMPLES. 
/.     Multiply  x^  by  x\ 


9 
10 
II 
12 
^3 


Multiply  x^  by  x^. 
Multiply  x^  by  x^. 
Multiply  —  jf*  by  x^. 
Multiply  —x^  by  —x^. 
Multiply  (—xy  by  (—x)\ 
Multiply  x^hy  (—xy. 
Multiply  x^  by  (—x^. 
Multiply  (\y  by  (\y. 
Multiply  (-\y  by  (-\y. 
Multiply  (lay  by  a\ 
Multiply  2*,  a^  and  a^  together. 
Multiply  (2ay  by  a^. 


Theory  of  Indices.  17 


ar 


i^.     Multiply  I  —  I     by  ;t:*. 

75.     Multiply  1^1    ,  ^^  and  a  together. 

f  I  1  ^ 
16.     Multiply     -        hy  x\ 

ly.  Multiply  x^hy  x^. 

18.  Multiply  x^  2^  and  x^  together. 

ig.  Multiply  x^  by  (2xy. 

20.  Multiply  3^  x^,  2^  and  x^  together. 

21.  Multiply  ^3-^/  by  (2xp. 

22.  Multiply  (2,^)^  by  (2x)^. 
2j.  Multiply  f  3-r/  by  (2x)^. 
2^.  Multiply  (^3-^/  by  (2x)^. 

2^.  Multiply  (2)^f  by  (2x)^.     Compare  with  example  2t. 

26.  Multiply  r  — 3-^/  by  (2x)^. 

2^.  Multiply  (x-\-y)^  by  (x-^y)\ 

28.  Multiply  (x—yy  by  (x—y)'-. 

2g.  Multiply  (x—yY  by  (x-\-y)'^. 

JO.  Multiply  (x^—y),  (x—y)^  and  (x'\-y)^  together. 

ji.  Find  the  value  of  [^ ("+'"' ]"+''. 

j2.  Distinguish  between  a^''^^  and  (a")^ 

7.  To  find  the  quotient  of  two  powers  of  the  same  quantity. 

a^     aaa 

a       aa 

aaaaa 

and  a^-^a?^=- z^aa^^d". 

aaa 

In  each  of  these  two  cases  the  quotient  is  seen  to  be  a  with  an 
exponent  equal  to  the  exponent  of  the  dividend  7ni7ius  the  ex- 
ponent of  the  divisor. 

aa       I 

Again,  a'-h-a^= =-, 

^  aaa      a 

aaa        i 
and  ^?3_j_^s__ = 

aaaaa     a^ 
In  each  of  these  two  cases  the  quotient  is  seen  to  be  a  fraction 
whose  numerator  is  i  and  whose  denominator  is  a  with  an  ex- 
ponent equal  to  the  exponent  of  the  divisor  vihius  the  exponent 
of  the  dividend. 


1 8  Algebra. 

Now,  in  general,   if  n  and  r  are  positive  whole  numbers  and 

a" _aaaa  .    .    .  to  n  factors, 
a''     aaa .    .    .    .  to  r  factors, 
and  the  r  a's  in  the  denominator  will  cancel  r  of  the  a's  in  the 
numerator  and  leave  (?i — r)  a's  in  the  iiuinerator. 

.' .  a"—a''=a"~''.  (c) 

This  formula  will  be  referred  to  as  formula  (c). 

Again,  n  and  r  still  being  positive  whole  numbers,  if  ^^<r,  we 

have,  as  before, 

a" _aaaa  .    .    .  to  n  factors, 

a''     aaa .    .    .    .  to  r  factors, 
but  here  the  n  a's  in  the  numerator  will  cancel  ?i  of  the  «'s  in  the 
denominator  and  leave  (r—7i)  a' s  in  the  denominator. 

^"-r-<2''=--^    when  n<^r.  (d) 

This  formula  will  be  referred  to  as  formula  (d). 

8.  We  may  also  find  the  quotient  of  the  same  powers  of  dif- 
ferent quantities. 

a^ _aa_  a  a _  f^]  ^ 

b^~  bb^  b  b^  \b\ 


and 

and  so  in  general, 


a^     aaa     aaa 
b'^JVb^l)  b  b 


a"     aaa  ...  to  7i  factors. 


b"     bbb  .    .    .  to  n  factors, 

: ,   ,   .  .    .    .  to  71  factors  each  of  which  equals  -. 
bbb  ^         b 


9.  In  the  case  just  considered  we  have  the  same  power,  but  the 
quantities  a  and  b  may  be  different,  but  in  Art.  7  we  have  the 
same  quantities,  but  the  powers  may  have  been  different.  The 
student  should  not  confuse  these  two  cases. 


Theory  of  Indices, 
examples. 


1.  Divide  x^  by  x^. 

2.  Divide  jt^  by  ;r5. 

J.  Divide  (—x)^  by  (—x)^. 

4..  Divide  — x^  by  —x^. 

5.  Divide  —  jf  ^  by  (—x)^. 

6.  Divide  —x'^  by  (—x)^. 

7.  Divide  (—x)^  by  (—x)\ 

S.  Divide  I -.'T''^'- 

by   ~| 


9.     Divide  I  — 


10.     Divide  (x—y)^  by 


-y) 


11.  Divide  (x—y)^  by  ,  , 

12.  Divide  (x^—y^)'^  by  (x—y)^. 

13.  Multiply  b^  by  3^  and  divide  the  product  by  b\ 

14.  Divide  b^  by  b''  and  multiply  the  quotient  by  b*. 
75.  Divide  b^  by  ^^  and  multiply  the  quotient  by  ^^ 

16.  Multiply  b^  by  ^^  and  divide  the  product  by  b\ 

17.  Divide  b^  by  b'^  and  multiply  the  quotient  by  b". 

18.  Divide  b"  by  If  and  multiply  the  quotient  by  b^. 
ig.  Multiply  b''  by  b"^  and  divide  the  product  by  ^. 

20.  Divide  b''  by  b^  and  multiply  the  quotient  by  b^. 

21.  Divide  b^  by  b^  and  multiply  the  quotient  by  b^. 

22.  Divide  c''  by  c^,  the  quotient  by  c^  and  so  on  until  five 
divisions  are  performed. 

2j.     Divide  c^  by  r'',  the  quotient  by  <:^  and  so  on  until  five 
divisions  are  performed. 

10.  We  saw  that  a"^a''=a"~''  i{  ?i^r.  Now  let  us  take  r=i, 
then  (n  being  a  positive  whole  number)  when  we  divide  a"  by  a 
we  simply  subtract  one  from  the  exponent. 

Now  let  us  take  some  number  for  ?i,  say  5,  and  divide  a^  by  a, 
the  quotient  by  a,  and  so  on  as  long  as  we  can,  each  time  per- 
forming the  division  by  subtracting  one  from  the  exponent. 


20         •  Algebra. 

We  obtain  the  following  equations  : 

If  we  attempt  to  go  one  step  further  by  the  same  rule,  viz :  sub- 
tract one  from  the  exponent,  we  get 

a-h-a=a°. 

Now,  a"  is  a  symbol  that  has  not  been  used  before,  and  indeed 
one  that  has  no  meaning  according  to  the  definition  already  given 
of  a  power  of  a  number.  But  we  know  that  a^a=i,  and  if  we 
agree  that  this  new  symbol  a""  shall  be  i,  <2  being  any  number 
whatever  (not  zero),  then  we  may  carry  our  process  of  successive 
division  one  step  further  than  we  could  without  this  agreement. 

More  than  this,  it  may  easily  be  seen  that  by  giving  this  mean- 
ing to  a°  each  of  our  formulas  (a),  (b),  (c),  (d)  is  slightly  more 
general  than  it  was  before.  Let  us  examine  these  formulas  sep- 
arately. 

11.  First,  a"a'-=a"-^\ 
If  we  here  make  ;/=o,  we  get 

a°a''=a'', 
and  this  is  true  if  a°=i. 

Again,  if  we  make  r=o,  we  get 

a"a°=a'\ 
and  this  is  true  if  a°=  i . 

12.  Second,  (a''y=a"''.  (d) 
If  we  here  make  n=o,  we  get 

which  is  true  if  «°=i,  since  i''=i  ;  and  if  we  make  r=o,  we  get 

and  this,  too,  is  true  if  afiy  quantity  affected  with  a  zero  exponent 
equals  one. 

*  The  student  may  think  that  such  an  equation  as 

40 2° 

involves  the  absurdity  that  4=2,  but  it  does  not.    No  one  thinks  that  the  equation 

4      2 

—=.—  involves  any  absurdity,  and  so  if  we  look  upon  a  quantity  affected  with  an  expon- 
ent zero  as  only  another  way  of  writing  a  quantity  divided  by  itself,  there  is  no  con- 
fusion. 


13.  Third,                       a"~a'-=a' 

H-r^ 

If  we  make  n=r  we  get 

a"-^a"= 

--a\ 

and  this  is  true  if  a°=  i . 

Again,  if  we  make  r=o  we  get 

a"-r-<2°= 

■  a", 

and  this  also  is  true  if  a°=  i . 

14.  Fourth,                       a"^a'-=^ 

I 

If  we  here  make  7i=r  we  get 

a"^a"= 

I 

and  this  is  true  if  «°=  i . 

Again,  if  72  =  0  we  get        a°~a''= 

I 

Theory  of  Indices.  21 


r-^; 


and  this,  too,  is  true  if  <2''=  i. 

15.  Now,  because  the  assumption  a°=\  leads  to  no  incoti- 
sistency  it  is  permissible,  and  because  it  gives  greater  generality 
to  our  formulas  it  is  advantageous. 

Therefore  we  adopt  the  equation  a°=  i  as  defining  the  meaning 
oia\ 

16.  The  question  naturally  arises,  is  there  any  way  whereby 
we  may  give  still  greater  generality  to  our  formulas  ? 

lyCt  us  look  again  at  our  process  of  successive  division. 
We  have  already  obtained  the  equations, 

a^-T-a—a\ 

a^-i-a  =  a^, 

a^^a=a^, 

a^-^a=-a, 

a-^a=a''=i. 
JTow,  if,  by  the  same  rule,  (viz :  subtract  one  each  time  from 
the  exponent,)  we  attempt  to  take  another  step  we  get 

a"-^a=a~\ 
Here,  again,  we  have  a  symbol  a~'  that  has  not  been  used  be- 


22  AI.GEBRA. 

fore,  and  one  which  has  no  meaning  according  to  the  definition  of 
a  power  of  a  number.     But  a"  being  i ,  we  know  that 

a 
and  so  the  equation, 

would  be  true  if  a~'  were  equal  to  -. 

a 

If  we  could  take  one  step  in  this  way  we  ought  to  be  able  to 

take  two  or  three  or,  indeed,  any  number,  and  if  we  could  do  this 

we  could  get,  in  addition  to  the  above,  the  following  equations : 

etc. 
As  we  have  seen,  the  Jirsf  of  these  equations  would  be  true  if 

«""=-,   and  from  this  the  second  would  be  true  if  a~^=     ,  and 
a  a 

from  this  the  third  would  be  true  if  a~^=—,  etc.,  and  the  set  of 
equations  just  written  might  be  carried  just  as  far  as  we  please  if 

q  being  any  whole  number. 

Let  us  examine  the  effect  of  this  supposition  on  our  formulas 

(a),  (b),  (c),  (d). 

17.  If  we  wish  the  quotient  a"H-«'' we  are  directed  in  Art.  7 
to  use  formula  (c)  if  ?^>r,  and  formula  (d)  if  n<,r. 
Suppose  «<r  and  r—71—q,  then  formula  (d)  gives 

a" 
But  if  we  should  try  to  use  formula  (c)  in  this  case  we  would 
get 

and  this  would  be  true  if 

so  that,  if  we  could  use  a  quantity  with  a  negative  exponent,  then 


Theory  of  Indices.  23 

formula  (c)  could  be  used  when  72<r  as  well  as  when  w>r,  and, 
if  we  like,  we  might  retain  formula  (c)  and  entirely  dispense  with 
formula  (d). 

Again,  it  may  be  seen,  in  a  similar  manner,  that  if  <^~''=— form- 

ula  (d)  could  be  used  when  /z>r  as  well  as  when  n<.r,  so  that 
we  might,  if  we  like,  retain  formula  (d)  and  entirely  dispense 
with  formula  (c). 

If  we  find  that  we  may  use  negative  exponents  upon  the  above 
interpretation,  then  we  will  for  the  most  part  dispense  with  form- 
ula (d),  using  it  only  now  and  then,  if  at  all,  when  it  comes  a 
little  handier  than  formula  (c). 

18.   Again,  by  the  above  interpretation  formula  (c)  can  be  used 
when  one  or  both  of  the  exponents  are  negative. 
First,  suppose  r  negative  and  equal  to  —q,  then 

a"^a-''=a"  -. —  =  a"a''= a"^". 
But  substituting  in  (c), 

the  same  result  as  before,  so  that  formula  (c)  may  be  used  when 


Second,  suppose  ?i  negative  and  equal  to  —s,  then 

a"  '  a'  a''     a' a''     a'^''' 

But  by  substituting  in  (c), 

the  same  result  as  before  if  our  interpretation  of  negative  ex- 
ponents be  correct,  so  that  formula  (c)  may  be  used  when  n  is 
negative. 

Third,  suppose  both  71  and  ^negative  and  let  /z=— ^and  r=— ^, 
then 


\       1     a 


a  '-^a  ^= — ; — =— =d;*  \ 
a      a''     a' 

But  by  substituting  in  the  formula, 

the  same  result  as  before,  hence  formula  (c)  may  be  used  when 
both  71  and  r  are  negative. 


24  Algebra. 

19.  Formula  (a)  niaj^  be  used  when  either  or  both  of  the  ex- 
ponents are  negative  if  the  above  interpretation  be  correct. 
First,  suppose  r=—q,  then 

But  by  substituting  in  the  formula, 

Second,  suppose  n=—q,  then 

«^ 
But  by  substituting  in  the  formula, 

so  that  formula  (a)  may  be  used  when  n  is  negative. 
Third,  suppose  ?i=—s  and  r=—q,  then 

-V  -^_  ^    ^  _    ^    _    ^ 
But  by  substituting  in  the  formula, 


^ 


+-?' 


r^"/    /.„w/  .«.- 


so    that    formula    (a)    may    be    used  where    both    n  and  r  are 
negative. 

20.  Formula  (b)  may  be  used  when  either  ?i  or  r  or  both  are 
negative. 

First,  suppose  r  negative  and  equal  to  —q,  then 

'     I     _  I 
(a")'~a^ 
But  by  substitution  in  the  formula, 

so  that  formula  (b)  may  be  used  when  r  is  negative. 
Second,  suppose  n  negative  and  equal  to  —q,  then 

But  by  substituting  in  the  formula, 

so  that  fonnula  (b)  may  be  used  when  7i  is  negative. 


Theory  of  Indices.  25 

Third,  suppose  both  exponents  are  negative  and  let  71=— s  and 
r=—q,  then 

a' 
But  by  substituting  in  the  formula, 

so  that  formula  (b)  may  be  used  when  both  exponents  are  neg- 
ative. 

21.  Thus  we  see  that  if  we  interpret  a~'^  as  being  - -,  a  being 

any  number  whatever  (not  zero),  and  q  being  any  whole  number, 
the  exponents  in  all  our  formulas  may  be  any  whole  numbers, 
positive  or  negative,  and  this  makes  our  formulas  considerably 
more  general  than  they  were  before. 

Now,  because  the  supposition  a~^^=~^  leads  to  no  inconsistency 

it  is  permissible,  and  because  it  gives  greater  generality  to  our 
formulas  it  is  advantageous. 

Therefore  we  adopt  the  equation  «~^=  —  as  defining  the  mean- 
ing of  a"'. 

22.  Since  <2~^=-  ,   and  therefore  -— ,=<3!^  it  follows  that  in 

a'^  a  ^ 

any  fraction  d^ny  factor  may  be  transferred  from  the  numerator  to 
the  denominator,  or  vice  versa,  by  simply  changing  the  sign  of  the 
exponent. 

Hence,  if  in  formula  (c)  we  transfer  «''  from  the  denominator  to 

the  numerator,   —  becomes  a"a~'',  which  by  formula  (a)  equals 
a'' 

^"-''•,  so  that  fonnulas  (a)  and  (c)  are  really  identical,  but,  for  the 

sake  of  convenience,  both  are  retained. 

A— 3 


26  Algebra. 

examples. 

„,  .      ab'  .  ,. 

7.     Write  -rr.  ni  one  line. 

TiJC^V" 

2.     Write      .    „  in  one  line. 

x'v^(a — by . 
?.     Wnte      "^  m  one  line. 

-^  I 


^.     Write         *^~    all  iii  the  loivcr  line. 


5".     Write-         ,    r- so  that  all  exponents  are  preceded  by 


the  +  sign. 


6.     Write  ^—-^ —  with  all  positive  exponents. 


23.  Having  now  dispensed  with  formula  (d)  and  extended 
formulas  (a),  (b),  (c)  so  that  the  exponents  may  be  a/iy  ivholc 
members,  positive  or  negative,  the  question  arises,  can  we  give  still 
greater  generality  to  our  formulas  by  using  exponents  which  are 
fractions  ? 

24.  If  quantities  with  fractional  exponents  have  an}-  meaning 
and  if  we  can  use  them  in  our  formulas,  we  must  have  by  form- 
ula (b) 


n  being  here  any  positive  zvhole  niunber;   i.  e.  a>'  is  a   quantity 
which  raised  to  the  ?ith  power  equals  a. 

Raise  both  sides  of  this  equation  to  the  rth  power,  r  being  a 
positive  whole  number,  and  we  get 

so  that,  if  we  are  permitted  to  use  fractional  exponents,  a>'  denotes 
the  ;'th  power  of  the  ;^th  root  of  a. 


Theory  of  Indices.  27 

25.  Again,    by    the    definition    of   a    quantity    with    an    ex- 
ponent —  I, 


and  by  formula  (b) 


U" '')-«" 


I 


or,  taking  the  reciprocal  of  both  sides, 

-'^       I 

a~': 

.26  Thus  we  have  suggestions  of  meanings  for  both  positive 
and  negative  fractional  exponents,  and  if  we  introduce  fractional 
exponents  into  our  formulas  with  the  meanings  suggested,  these 
formulas  will  be  found  to  give  consistent  results,  as  we  shall  see. 

27.  Before  substituting  in  our  formulas  it  is  necessar>'  to  stop 
and  show  that,  with  the  meanings  suggested,  a  quantity  with  a 
fractional  exponent  has  the  same  value  whether  the  exponent  is 

in  its  lowest  terms  or  not. 

1 
Let  «^"=JL- 

then  ^=jt-^"=^ji-?;" 

i_ 

In  a  similar  manner  it  may  be  shown  that 

_ ''        _'/'" 
a    "—a  f" 


28.  Examination  of  formula  (a). 

r        .  p  . 
Let  —  and  -  be  any  two  positive 
n  q 

two  negative  fractions.     Then  there  are  four  cases  to  consider. 


Let-  and  -  be  any  two  positive  fractions   and and 

n  q  ^  ^  n  q 


28  Algebra. 


FiJ'st  case. 


L.    L 
First,  a"a'^=  what? 

Second,  a" a    '^  =  what? 

Third,  a~^a^=  what? 

Fourth,  a   "a   '^=  what? 

a"a'^  =a"fa'"^  by  art.  27. 


and  by  direct  substitution  in  the  formula  we  get 

a"  a'^  =a"^  'f 
Second  case. 

a" a    '/  =  a"''a  '"!=  j <2"'/       a'"^ ] 

and  b}^  direct  substitution  in  th?  formula  wc  also  get 

a" a    '^=a"    '' 
Third  case. 

This  is  the  same  as  the  second  case,  only  the  order  of  the  fac- 
tors is  changed,  and  therefore  as  in  the  second  case  the  result  will 
be  the  same  as  given  by  direct  substitution  in  the  formula. 

Fourth  case. 

--^-^11  I 

a    "a     ''=    -v  -/=    ,    7 

a''  a'^      a"a'f 

I  -(-  +  ^)       ----^    • 

a"^  'i 
, V  ^"^  ^^y  direct  substitution  in  the  formula  we  also  get 

Thus  we  see  that  by  using  fractional  exJponents  according  to 
the  suggestions  before  obtained,  the  result  of  multiplying  two 
fractional  powers  of  a  is,  in  ever}^  case,  in  perfect  accord  with  for- 
mula (a). 


Theory  of  Indices. 


29 


29.  Examination  of  formula  (b). 

As  before,  let  -  and  -    be   any    two    positive    fractions,    and 
n  q 

— -  and  —  -  any  two  negative  fractions.     Then  there  are  four 
n  q 

cases  to  consider. 

( 


First,   k"  "'=  what? 


First  case. 


Second,    [«"  )    ^=  what? 
Third,    \a    "  \' =  what? 
Fourth,   {a    "\     '=  what? 


Let 


or    {a"\  '^x'-'; 
From  ('i  j  «"'.==jt-''^; 


(2) 
(3) 

(4) 
(5) 

((>) 
(1) 


.-.  from  (6)  and  (-])  \a"  J  '=fl«^ 
and  by  direct  substitution  in  the  formula  we  also  get 


Second  case. 


rp 


a"\     " 


/—      rp—^       "^' 


a"  Mi'      a"f 


and  bv  direct  substitution  in  the  formula  we  also  get 


Third  case. 


a"\     '=a 


1  1 


I- 


9      a"f 


30 


AI.GEBRA. 


and  by  direct  substitution  in  the  formula  we  also  get 


Fourth  case. 


.    I 

I 

I 

I 

r 

"^ 

a« 

a" 

and  by  direct  substitution  in  the  formula  we  also  get 


'z 
■=a'"J. 


Thus  we  see  that  by  using  fractional  exponents  according  to 
the  suggestion  before  obtained,  the  result  of  raising  any  frac- 
tional power  of  a  to  any  other  fractional  power  is,  in  every  case,  in 
perfect  accord  with  formula  (b). 

30.  Examination  of  formula  (c). 

As  before,  let  -   and    -  be  any  two  positive  fractions  and  — 
n  Q  '^ 

P 
and  —     any  two  negative  fractions.     Then  we  have  four  cases  to 

consider. 

First,  a"  -^a'l  =  what  ? 


Second,  a"  -^a 


what  ? 


Third,  <2    " -r-^! '-^  =  what  ? 


First  case. 


Fourth,  a    '•  —a    "  = 
a"  -—a'^  =a" a    '' =a" 


what  ? 


by  Art.  29,  second  case,  and  by  direct  substitution  in  the  formula 

we  also  get 

1       /        /-  _  / 
a"  -7-a'^  =a"      '' . 
Second  case. 

^         _  A         !L    t.         ^  •  Z 
a'' -—a    9—a''af=a"^'f, 

and  by  direct  substitution  in  the  formula  we  also  get 
a"  -r-a    ''^=<2"  ^  ^. 


Theory  of  Indices.  31 

Third  case. 

_  ^-  /  _  ';    _  /  _  L  _  L 

a    "-^af=a    "a    ^=^    "      '^ , 

and  by  direct  substitution  in  the  formula  we  also  get 

_  Ji       t       _  1  _  z*^ 
a    "—a"— a    "      ^. 

Fourth  case. 

a    ''- -^a    '''=«    "af=a    "      f  ^ 
and  by  direct  substitution  in  the  formula  we  also  get 

_  -!L         _  A  _  il  4.  j^ 

a    "--a    '^  ==a    "      f. 
Thus  we  see  that  by  using  fractional  exponents  according  to 
the  suggestion  before  obtained,  the  result  of  dividing  one  frac- 
tional power  of  a  by  another  fractional  power  of  a  is,  in  every 
case,  in  perfect  accord  with  formula  (c). 

31.  Now,  because  the  suppositions 

a"  ={ Va Y  and  a    "  =  --^ :_ - 

lead  to  no  inconsistency  they  are  admissible,  and  because  they 
give  greater  generality  to  our  formulas  they  are  advantageous. 
Therefore  we  adopt  these  equations  to  define  the  meaning  of  quan- 
tities affected  with  fractional  exponents. 

32,  The  formulas  (a),  ( b),  (c)  are  now  so  generalized  by  the 
above  definitions  that  they  can  be  used  when  the  exponents  are 
any  positive  or  negative  whole  numbers  or  fractions,  and  it  might 
naturally  be  asked,  is  this  the  greatest  generality  of  which  they 
are  capable  ? 

•  Excluding  the  so-called  imaginaries,  there  is  no  kind  of  alge- 
braic numbers  not  yet  discussed  except  incommensurable  num- 
bers, and  the  consideration  of  quantities  affected  with  incommen- 
surable indices  is  reserv^ed  for  chapter  XI.  In  the  meantime, 
however,  it  should  be  remembered  that  the  formulas  are  to  be 
used  only  when  the  indices  are  commensurable. 


32 


Algebra. 


33.  By  means  of  the  meanings  now  given  to  negative  and 

fractional  exponents  it  is  easy  to  see  that  the  formula 

a"b"c"  .    .    .  =(a  b  c  .    .    .  )" 

holds  whether  ;/  is  positive  or  negative,  integral  or  fractional. 

I  —  ^ 

First,  let  ?/=    ,  a  positive  fraction,  and  let  a''=-x  and  b'' =^y\ 
r 

.-.  a=^x"  and  b=^y'' 
1    _i_ 
then  a'' b''  =^xy, 

and  ab=x''y''=^(xy)'\ 


1 


.-.  (ab)'-=xy 
y    \  1 

^  .-.  a'' b''  =  (ab)'' . 

1 
Multiply  both  sides  by  c  and  we  get 

L   1  A  11  A 

a  '■  b'' C  =(ab)''c''  =  (abc)  '' , 

and  so  on,  evidently,  for  any  number  of  factors. 

This  is  quite  an  important  formula,  stated  in  words  it  is, 
The  product  of  the  r  th  roots  of  several  quantities  equals  the  rth 
foot  of  their  product. 

Second,  let  7i=—r,  a  negative  quantity,  either  integral  or  frac- 
tional, then 

a-'b-'=~  \^=    \^  =  (abr\ 
a"^  b      (ab) 

Similarly  a~''b~''c~''=  (abc)~'\ 

and  so  on,  evidently,  for  any  number  of  factors. 

34.  The  formula   7„=='  t[  also  holds  good  whether  7z  is  posi- 
tive or  negative,  integral  or  fractional. 

1  1  1 

First,  let  n=j^,  a  positive  fraction,  then    --=^'-^-1    =  -]-  . 
.. -^  ,  —  yb )  ib ) 

0'' 

Stated  in  words,  this  is. 

The  quotierit  of  the  rth  roots  of  two  qiiayitities  equals  the  rth  root 
of  their  quotieyit. 


b-"     a"     \a\       \b 


Theory  of  Indices.  33 

Second,  let  ?i=—r,  a  negative  quantity,  either  integral  or  frac- 
tional, then 

a-''     b'-     \bV     wy     \a 
\b 

EXAMPLES. 

1.  Write   the   following   expressions,   using   fractional  ex- 
ponents in  place  of  the  radical  signs  : 

^^  a\x  \jlry  >J">  ' 

2.  Write  the  following  expressions,  u.sing  radical  signs  in 
place  of  fractional  exponents : 

. .  5  jt^ 

J.     Multiply  ^    ^  by  Jtr'-^. 

1  _  .1 

4.  Multiply  x'^  by  Jt:    '\ 

5.  Multiply  -^  by  -^. 

x^        X   ^ 

6.  Multiply  [ip'^by-^. 


(xj 


n 
\T"(J 


7.  Divide  (x+yjUhxCx-^j')''^- 

8.  Divide  M^xy'+x^y~y^  by  x^—y 
p.     Divide  jf^-fy^  by  x'- -\-y" . 


1-  L  ?L  L  \. 

10.  Multiply  X-  —x'^ -\-x'' -x"- -^x- -x-^x"- -\  byy^'+i. 

11.  Simplify   Xx'^-^x']  '^. 

12.  Simplify   (.r^jz-'^-^J  . 

x^y~'2^. 
i/f..     Find  the  continued   product  of 


34  Algebra. 


75.     Multiply  Jf  ^  —X     -    by  [jt"^— Jt-    ^J 

16.     Simplify    [{J^^}"']^' 

_  j_ 

(  x''"  )    " 
ly;     Simplify  ^. 

.3.  _i         i 

18.     Simplify  —^ ^ { , 


/p.     Sii 


^.       ,.^     2>('2x)^(%bx)^ 
20.     Simplify ^/  ^         ' 

(ax)'^\^  6a 


CHAPTER  III. 

RADICAI.   QUANTITIES   AND   IRRATIONAI,   EXPRESSIONS. 

I.  From  the  last  chapter  the  student  has  learned  that  there  are 
two  methods  in  use  for  indicating  the  root  of  a  quantity,  one  by 
the  ordinary  radical  sign  and  the  other  by  a  fractional  exponent. 
Of  course  it  is  entirely  unnecessary  to  have  two  modes  of  writing 
the  same  thing,  and  in  this  sense  either  one  of  the  two  ways  may 
be  considered  superfluous.  But  practically  each  method  of  nota- 
tion has  an  advantage  in  special  cases,  and  the  student  will  feel 
this  as  he  proceeds.  This  fact  that  both  methods  are  better  than 
either  one,  accounts  for  the  retention  of  both  in  mathematics. 

2.  HiSTORiCAii  Note— The  introduction  of  the  present  symbols  into  alge- 
bra was  very  gradual,  and  the  use  of  a  particular  symbol  did  not  generally 
become  common  until  some  time  after  its  suggestion.  The  signs  -f-  ^^^^  — 
were  first  used  at  the  beginning  of  the  16th  century  in  the  works  of  Gramma- 
teus,  Kudolf  and  Stifel.  Recarde  (born  about  1500)  is  said  to  have  invented 
the  sign  of  equality  about  this  time.  Scheubet's  work  (1552)  is  the  first  one 
containing  the  sign  ^Z  .  and  Vieta  (born  1540)  first  used  the  vinculum  in  con- 
nection with  it.  Before  this,  root- extraction  was  indicated  by  a  symbol  some- 
thing like  I]^. /XStfixLn  (born  1548)  first  used  numbers  to  indicate  powers  of  a 
quantity,  and  lic-even  suggested  the  use  of  fractional  exponents,  but  not  until 
Descartes  (born  1596)  did  exponents  take  the  modern  form  of  a  superior  figure. 

The  development  of  the  general  notion  of  an  exponent  (negative,  fractional, 
incommensurable)  first  appears  in  a  work  of  John  Wallis  (pubUshed  in  1665) 
in  connection  with  the  quadrature  of  plane  curves. 

To  show  the  appearance  of  mathematical  works  before  the  introduction  c;f 
the  common  symbols,  we  give  the  following  expression  taken  from  Cardan's 
works  (1545) : 

1^  V.  cu.  1^  108  p7  10    I    mj^  cu.  J^  108  m  10, 
which  is  an  abbreviation  for  "Radix  universalis  oubica  radicis  ex  108  plus  10, 
minus  radice  universali  cubica  radicis  ex  108  minus  10."     Or,  in  modern  sym- 
bols, ■ 

^>/i'o8-fio-^^^io8~io 
Here  is  a  sentence  from  Vieta's  work  (1615). 
Et  omnibus  perE  cubum  ductis  et  ex  arte  concinnatus, 

E  cubi  quad,  -f  Z  solido  2  in  E  cubum,  acquabitur  B  plani  cubo. 
This  translated  reads  :     Multiplying  both  members  ("  all  ")  by  K-t  ami  imit- 
ing  like  terms, 


36  AI.GEBRA. 

3.  Definitions.  In  the  following  pages,  by  the  word  Radical 
may  be  understood  the  indicated  root  of  an  expression,  whether 
that  root  is  indicated  by  the  ordinary  radical  sign  or  by  a  frac- 
tional exponent. 

By  the  Index  of  a  radical  may  be  understood  either  the  number 
written  in  the  angle  of  the  radical  sign  or  the  denominator  of  the 
fractional  exponent. 

A  multiplier  written  before  a  radical  will  sometimes  be  called 
the  co-efficient  of  the  radical. 

A  Simple  radical  is  the  indicated  root  of  a  rational  expression. 

A  Complex  radical  is  the  indicated  root  of  an  irrational 
expression. 

A  monomial  Surd  is  the  name  applied  to  the  indicated  root  of 
a  commensurable  number,  when  that  root  cannot  be  exactly  taken; 
as  x/|,  or  V3- 

If  all  the  irrational  terms  in  a  binomial  or  polynomial  are  surds, 
it  is  called  a  binomial  or  polynomial  surd,  as  the  case  may  be. 

It  should  bo  noticed  here  that  we  make  a  distinction  between  the  terms 
irrational  expression  and  surd,  a  distinction  which  is  not  commonly  made, 
the  two  terms  being  generally  defined  as  identical.  According  to  the  above 
definition,  ^4  ^'^-\-\/t  ^^l3,  '^ ~  are  not  surds.  But  they  are  irra- 
tional by  the  definition  of  I,  Art.  3,  This  limited  meaning  of  the  word  surd  is 
convenient  and  is  growing  in  use.  It  is  found  in  both  Aldis'  and  Chrystal's 
algebras. 

Radicals  are  said  to  be  Similar  when  they  have  the  same  index 
and  the  expressions  under  the  radical  signs  are  the  same ;  that  is, 
two  radicals  are  similar  when  they  differ  only  in  their  coefficients. 
Such  are  ^\^ ab  and  ms/ ab;  also  f^y  and  f^y. 

4,  Definition.  For  a  radical  to  be  in  its  siinplest  form  it  is 
necessary  (i)  that  no  factor  of  the  expression  under  the  radical 
sign  is  a  perfect  power  of  the  required  root ;  (2)  that  the  expres- 
sion under  the  radical  sign  is  integral ;  (3)  that  the  index  of  the 
radical  is  the  smallest  possible. 

It  will  be  seen  from  the  following  pages  that  every  simple  radi- 
cal can  be  placed  in  this  form  without  changing  its  value.  The 
transpositions  necessary  to  effect  the  reductions  depend  upon  cer- 
tain principles,  or  theorems,  established  in  the  last  chapter,  which 
we  collect  here  for  reference. 


Radicals.  37 

5.  The  71  th  root  of  the  product  of  several  quantities  is  equal  to  the 
product  of  the  71  th  roots  of  the  several  quantities. 

That  is,  VabF^'.=Va  Vl  V~c  .    . 

1       A  I.  Jl 
or  (a  b  c  .    .  )''=a" b"c"  .    . 

6.  The  71  th  root  of  the  quotient  of  tivo  7iumbers  is  equal  to  the 
quotient  of  their  71  th  7vots. 


That  is,  __     „ 


i_       1 
f<2 )  "      a 
or 


Sb^^Tb 

l_       1 
\aY  _a" 

ib)     -   -L 


b> 

7.    The  7irth  root  of  a  qua7itity  equals  the  11  th  I'oot  of  the  rth 
root  of  the  qua7itity. 

That  is,  V^=Vv^, 


or 


(ay-^\(a)^\ 


8.  To  REMOVE  A  Factor  from  beneath  the  Radical 
Sign.  When  any  factor  of  the  quantity  beneath  the  radical  sign 
is  an  exact  power  of  the  indicated  root,  the  root  of  that  factor  may 
be  taken  and  written  as  a  coefficient  while  the  other  factors  are 
left  beneath  the  radical  sign.  Thus  v/128  may  be  written  v/64  X  2, 
which,  by  iVrt.  5,  equals  v^64X  v^2,  which  equals  8v/2.  As  an- 
other case  take  "^ \6ax\  which  equals  ^8^^ X  2ax=  ^8-r^ X  -^ 2ax 
—  2  x\^^2ax.  It  is  readily  seen  that  this  same  process  may  be  ap- 
plied to  any  similar  case. 

9.  Examples.  Remove  as  many  factors  as  possible  from  be- 
neath the  radical  signs  in  the  following : 

W50. 


x/8io. 
^87o. 

\\^']2x^y. 
v^'^2808. 


38  AlvGEBRA. 

p.     -     >/ aj(f -\- 2X^ . 


10.      \^iSa^b^ 


II.      V  192^^-rj}/" 


12.     V(a—b)"(b-\-a)''^\ 

10.  ^o  Intkgralize  the  Expression  under  the  Radicae 
Sign.     Suppose  we  wish  to  transform  the  radical 

3  lab"" 

sjxf 
so  that  there  shall  be  no  fraction  under  the  radical  sign.     Multi- 
ply both  numerator  and  denominator  of  the  fraction  by  a  quantity 
that  will  render  the  denominator  a  perfect  cube,  thus  : 


3  M'_3  \ab' 


x^y__z  \ab^x^y 
x^y     ^    x^y^ 


But,  by  Art,  6, 


siab'xy     ^'ab'xy       i      , 

-  =      ,^^ab\vy. 


In  general,  to  integralize  a  radical  of  the  form     j-,   multipl 
numerator  and  denominator  by  b"~^  and  we  obtain 


n  \ab"-' 

^ 

w> ' 

which,  by  Art.  6,  equals 

Vaiy- 

V^« 

and  this  is  equal  to 

iv..»-, 

which  is  in  the  required  form. 


Radicals. 


39 


II.  Examples.  Integralize  the  expressions  under  the"  radicla 
signs  in  the  following,  simplifying  the  result  in  each  case  by  Art. 
8,  if  necessary: 


I. 


Process 


V.47     N49X3~W49X9~^'''^^- 


4 
5 
6 

7 
8 

9 
10. 


(a-b)"- 


12.  To  LOWER  THE  Index  OF  A  RADICAL.     It  is  plain  that 
V'25  by  Art.  7  =  n/~^ 


4/ 


25- 


>/  5  ;  also  that  V  ^a^=\^ ^'7~^^=^\/2a; 

similarly  ^y ^b^=^^  s/ Ab^'="^^ 2b  ]  and  in  general,  "v^<2"='^^V'^== 

''^a .  From  this  we  see  that  the  index  of  a  radical  can  be  lowered 
if  the  expression  under  the  radical  is  a  perfect  power  correspond- 
ing to  some  factor  of  the  original  index. 

13.  Examples.     Reduce  the  following  to  their  simplest  fonns. 
See  Art.  4. 

I.      >/  '^^x'y^. 

6/ „ 


3' 
4- 


1000 


9  1^ 


40  AI.GEBRA. 


6. 


gx^—iSxj-i-gj''' 


7.     2^a—d—V64d'-\-64a'—i28ad. 

14^     To    INTRODUCE    A    COEFFICIENT    UNDER    THE     RadICAI^ 

Sign.  It  is  sometimes  convenient  to  have  a  radical  in  a  form 
without  a  coefficient.  The  coefficient  can  always  be  introduced 
under  the  radical  sign  by  the  inverse  of  the  method  of  Art.  8. 
Thus,  2X'^ 2ax=^^ S x^"^  2ax—y^^ i6ax* ;  similarly,  a'\^  c=V a"c. 

15.  Examples.     Place  the  coefficients  in  the  following  under 
the  radical  sign  without  changing  the  value  of  the  expression  : 

I.  TyttX^s/  2)CIX. 


x^  a—x. 

50^50- 

a-b^\^  X  — 


y- 


16.  Addition  and  Subtraction  of  Radicals.  Similar  rad- 
icals (Art.  4)  can  be  combined  by  addition  or  subtraction ;  and  if 
they  are  dissimilar  no  combination  can  take  place.  Take  for  ex- 
ample the  expression, 


V^^'+2v/yV-2j^— 4-^^IO. 


Reducing  each  expression  to  its  simplest  form,  it  becomes 
ax'''^  T^a-\-\^  \o—^ax^\^  2f^-\-\^  10. 
It  is  now  noticed  that  the  first  and  third  and  the  second  and 
fourth  radicals  are  similar  to  each  other ;  whence,  grouping  sim- 
ilar terms,  the  expression  becomes 

(ax'-^ax^-)^2>^  +  (i+\)^/io, 
or  \ax''V 2)a-\-i\'>/ 10. 
We  observe  here  the  necessity  of  reducing  each  of  the  radicals  in 
any  given  expression  to  its  simplest  form,  for  then  it  can  be 
told  whether  or  not  any  number  of  the  radicals  are  similar  to 
each  other  and  consequently  whether  or  not  they  can  be  combined 
together. 


Radicals.  41 

17.  Examples.     Give  the  value  of  each  of  the  following  ex- 
pressions in  as  simple  a  form  as  possible  : 
/.      lov^f-f  V  1000. 

3.     2  n/48  +  3^  147-5^75- 

4..      v^98  +  'v^72  +  'v/ 242. 

5-      ^7^*+  ^^b^-\-  \^4Sa'd\ 


6.     |v4ox=— 3V625Jt:^+io^500oJr". 

^  •         5  ^     5^8^147         2  ^    ^• 


P- 


JO.     Prove    h+^^-^+    Igfl+^^+^^^^Jgl+j-U 


X 


18.  Multiplication  and  Division  op  Radicals.  The  pro- 
duct of  several  radicals  of  the  same  index  may  be  expressed  as  a 
single  radical  by  means  of  Art.  5.     Thus 

^^2X  \/3X  V ^  —  V 2  X3X5=v^30  ; 
^r^r^X  ^ mx X  ^V^i=^^ x^ r^frf=rx^rnf  ; 
VaxVbxVc.    .    .—Vabc.    .    . 
The  result  should  always  be  reduced  to  its  simplest  form.     If 
there  are  coefficients  they  should  be  multiplied  together  for  a  new 
coefficient,  for 

ay^^x  b'y/y  c^/ z=-abc>/ x  Vj/    'V z=^abc"y xyz. 
The  quotient  of  one  radical  by  another  of  the  same  index  may 
be  expressed  as  a  single  radical  by  means  of  Art.  6.     Thus 


Vb     sjb     b 

The  result  should  always  be  expressed  in  its  simplest  form. 

If  we  wish  'to  multiply  or  divide  radicals  of  different  indices  we 

must  first  reduce  them  to  a  common  index.    This  can  be  done  by 
A— 5 


42  Algebra. 

expressing  the  radical  by  means  of  fractional  exponents  and  then 
reducing  them  to  a  common  denominator,     Thus 

=  4"^^  3^'^-  2TV  (by  ^^j-^  8)='^7'X3'X2'=2'i/io8. 

ji_  

Also  ^  =  5;^_i.f5^_,i^-^,o. 

\J^3     ^T-w       ^3"      " 

19.  Examples.     Find  the  value  of  each  of  the  following  ex- 
pressions : 

/.  \^  T,acX  ^y  2a7n  X  "^bax. 

2.  N/|xx/fXv/i. 

3.  N/iX^'^f 

4.  a'^y  a — xy.x^s/  a-\-x. 

5-       ^  TO  X  ^    TT- 

/5.      2x'^  X  3X^. 


7- 


'"Nf^^""Ji 


S.  ^24X6^3. 

7^0.  2^/ 6-f-6'v^  2. 

Z^.  -V^  2  -7-^/2. 

ij.  ^  a'—x^-^^  a—x. 


J5-     5-^^3- 


7<5. 


v^fl5 — X  \^a-\-x 
A  polynomial  involving  radicals  is  generally  more  easily  multi- 
plied or  divided  by  another  such  polynomial  by  first  expressing 
the  radicals  by  fractional  exponents.  As  shown  in  Chapter  II, 
the  work  will  then  be  no  different  in  principle  from  the  case  when 
the  exponents  are  integral.  But  in  a  few  of  the  simpler  instances 
it  is  unnecessary^  to  pass  to  fractional  exponents,  e.  g. 


RaDICAIvS. 


43 


I  J.      Multiply  3  — \/6  by  v/ 2—^/3. 
Process:  3—    v/6 

v/2—   v^3 
3V/2  — 2V/3 

-3^3  +  3^2 
6v/2~5v/3 
Find  the  value  of 

18.  (v/3-v/2)(>/3  +  n/2). 

i<p.  (2'v^5  — 3v^2  +  v/ io)(v^i5  — \/6). 

20.  (vV4_v/6)(3^6  +  4^/3). 

21.  (3^^  45-7^5X^^+2-/-^" 


Result. 


•^'9i)- 


20.  Powers  and  RooTvS  of  Radicals.  It  has  already  been 
shown  (Chapter  II)  that  we  can  raise  a  quantity  affected  by  a 
fractional  exponent  to  any  required  power  by  multiplying  the 
fractional  exponent  by  the  exponent  of  the  required  power.  It 
was  also  shown  that  any  root  of  a  quantity  affected  with  a  frac- 
tional exponent  could  be  found  by  dividing  the  fractional  ex- 
ponent by  the  index  of  the  required  root.  Hence  we  can  find 
any  power  or  any  root  of  a  radical  if  it  is  expressed  by  means  of 
fractional  exponents ;  but  of  course  in  the  simpler  cases  the  con- 
venience of  fractional  exponents  will  not  be  felt.  We  give  one  or 
two  illustrations  and  leave  the  student  to  his  own  method  ;  the 
chief  requirement  is  that  he  should  be  able  to  show  that  his  work 
is  established  on  sound  principles. 

The  result  should  appear  in  its  simplest  form. 

21.  Examples. 

1.  Square  |^a^. 

Process:     {\^~CL^Y=^%[^^~a'y^%^  a'=-%a^^~a. 

2.  Find  the  fourth  power  of  ^v^/12. 


J.     Cube 


Sa' 


5- 
6. 


Square  3"^  3. 
Square  \^  2. 
Cube  a.x"^  ax. 


44  Algebra. 


7.  Find  the  value  of 

8.  Find  the  value  of 


y 


a"—b" 

x"'~'' 


^a+x 

g.     Find  the  cube  root  of  |^3. 
07ie  process:    ^fv/'3=^|^v/3  (Art.  5)  =  v''^|  V3,   (Art.  7) 

=  V|fx3(Art.  i8)  =  2^J^=2x/-^=tv/3. 
-Ajiothcr process;        -|'^3  =  '^ff'  (Art.  13) 

10.     Find  the  fourth  root  of  ypjf-v^'^gjr'y^ 
z/.     Take  the  cube  root  of  3'^3. 
12.     Take  the  cube  root  of  s/\, 

7?.     Find  the  value  of  '  I-    |£ 

7^.     Simplify  J.9^, 

^  98)'! 
75.     Find  the  value  of  (f  v^'^p'  x  v''^  |v^. 

...  Simplify  ;)"^;S 

22.  Rationalizing  Factors.  Any  multiplier  which,  when 
applied  to  an  irrational  polynomial,  will  free  it  from  radicals  is 
called  a  Rationalizing  Factor.  Thus  5  — "^3  is  a  rationalizing 
factor  for  the  binomial  surd  5 +  '^3,  since  (5-f  \/3)(5  — \/3)  = 
25—3=22.     Similarly   the   rationalizing   factor  for   ^3  — "^5    is 

^2>-^^b^  since  (^3  — ^5)(^34-^^5)  =  3-'  3—^- ^'^~ 

For  larger  polynomials  it  may  be  that  the  ratfonalizing  factor 
is  itself  composed  of  several  factors.  Take  the  quadratic  tri- 
nomiai  surd  ^^2  — ■v^3./  The  rationalizino;  factor  is 


/^ 


(v/2  — n/3  +  v/7)(2  — 2v/6)for 
(v/2 — v/3  +  \/7)(\/2  — v/3  — v/7)(2  — 2'V^6) 

=  |(v/2-v/3r-(N/7T](2-2N/6) 
=  — (2H-2V^6)(2  — 2V^6)=20. 


Radicals.  45 

23.  Problem.  To  Rationalize  any  Binomial  Quadratic 
Surd.  Any  binomial  quadratic  surd  may  be  represented  by 
a^p-^b^q,  where  a  and  b  may  be  either  positive  or  negative. 
The  rationalizing  factor  is  plainly  a\^p—b\^q,  for 

{a\^p  +  bs/q){a\^p—bs^~q)=^a-p—b'q^ 
which  is  rational. 

24.  Problem.  To  Rationalize  any  Trinomial  Quad- 
ratic Surd.  Any  trinomial  quadratic  surd  may  be  represented 
by  a^p-\-b^q-\-c'^r,  where  a,  b,  and  c  are  supposed  to  be  any 
rational  quantities  whatever,  positive  or  negative  or  integral  or 
fractional.     Multiply  first  by  ay^p-\-b\^q—c\^r,  and  we  obtain 

(a^p-^b\^q-\-cs/'r){a-/p^b\^q—c\^'r)  =  {a^p  +  b^~qy—{c\^rY 
=a'p-\-b^q—c^r-\-2ab^pq,  (i) 

which  is  rational  as  far  as  r  is  concerned.     Now  multiply  this  by 

{a-p-\-b~q—c'^r)^2abs/pq  (2) 

and  we  obtain 

{{a'p-\-lfq—c^r)-\-2ab>'^pq\{{a'p-\-b''q—c^r)  —  2abyjq\ 

=  (^arp-Vb^q-c^rXf-_^a^b^pq,  (3) 

which  is  rational  with  respect  to  all  the  quantities.  The  ration- 
alizing factor  for  the  original  trinomial  quadratic  surd  is  thus 
seen  to  be 

{a  yp  +  byq—c^l^){a'p-\-b'q—c''r—2ab^p}^  (4) 

25.  The  second  parenthesis  in  (4)  above  will  be  found  to  be 
composed  of  the  two  factors 

(aVp—b'yq-^c^r)(a\^p—b\^q—c\^r) 
Hence  the  rationalizing  factor  of  aVp  -{-b\/jif^C\/^  may  be 
written 
(aV])  J^bV'q  —cs/ r )(as^p—bV q  -{-%^/r_)(a>/p—b^q  —c>^r ) 
Observe  that  the  terms  of  each  of  the  component  trinomial  fac- 
tors of  this  expression  are  those  of  the  given  irrational  quantity 
and  the  signs  are  those  exhibited  in  the  scheme — 

+  -h  - 
+  -  + 
+ 


^<--'^ 


46  Algebra. 

Now  it  is  evident  that,  keeping  the  first  sign  unchanged,  there 
is  no  other  arrangement  of  signs  than  those  written  in  this  scheme, 
except  the  arrangement  +  +  + ,  which  is  the  arrangement  of  the 
given  trinomial.     Therefore 

The  ratio7ializing  factor  for  any  trinomi7ial  quadratic  surd  is  the 
product  of  all  the  different  trinomials  which  caii  be  made  from  the 
original  by  keeping  the  first  term  unchanged  a?id  giving  the  sig?is 
+  and  —  to  all  the  remai7iing  terms  in  every  possible  order,  except 
the  order  occurri?ig  in  the  given  tri?iomial. 

As  an  example,  find  the  rationalizing  factor  for  ^S~^7~^'^3- 
The  above  method  shows  it  to  be 

and  multiplying  the  original  trinomial  by  this  the  rationalized  re- 
sult is  found  to  be  —40. 

The  above  problem  is  cabable  of  generalization,  but  its  proof 
cannot  be  practically  given  here.  The  generalized  statement  is 
as  follows: 

The  ratiofializing  factor  for  any  polynomial  quadratic  surd  is 
the  product  of  all  the  differe7it  polyjiomials  which  ca7i  be  made  from 
the  origi7ial  by  keepi7ig  the  first  teri7i  imchanged  a7id  givi7ig  the 
sig7is  -j-  a7id  —  to  all  the  re77iai?ii7ig  terms  in  every  possible  07'der 
except  the  order  occurring  i7i  the  give7i  polyno77iial. 

26.  Problem.    To  Rationalize  any  Binomial  Surd.     A  bi- 

nomonial  surd  will  either  take  the  form  a^r-^c~q  ox  a  r—  cl.  Now 
since  these  fractional  exponents  may  be  reduced  to  a  common  de- 

nominator  so  that  the  expressions  become  a^<i-\-c''i  or  a'-^^—C'^  these 

.V  t  s  t 

binominal  surds  may  be  supposed  in  the  form  a''-\-  c"  ox  a"  —  c". 
These,  then,  are  the  only  forms  necessary  to  consider,  since  all 
binomial  surds  are  reducible  thereto. 


(a)    To  7'atio7ialize  the  fo7'7n 


a"  —  c 


For  convenience   let  a"=x   and  <:"=j';when  a"  —  c"  =^x—y. 
Now  multiply  x—y  by 

x"-'-^x"-y+x"-Y-\-  .    .    .y"-'  (i) 

and  we  obtain 


Radicals.  •  47 

(jt--j)(x"-'+jt-"->-fx"-y+  .   .   ,y-^)=x"-y\ 
by  substitution,  I,  Art.  26.     But 

which  is  rational.     Therefore  (i)  is  "the  rationalizing  factor  for 

a"  —  c" . 

jL        J 
(b)    To  rationalize  the  form  a"  -\-  c" . 

—  -L.  1        L 

As    before,     let    a"^=x    and     c"=^y\    whence   a"  -\-  c"  =^x-\-y. 

Multiply  x-\-y  by      ♦ 

x"-'—x"-'y-\-x"-^y—   .    .    .  zhy"-\  (2) 

The  product  is 

f  x''—y"  if  71  is  even, 

( .r"+y  if  wisodd, 


(x-\-y)(x"-^-x"-y-\-x"-y-  . 

■  -^y-'^^x. 

by  I,  Arts.  27,  28.     But 

■^'  —y  =  \a" 

{       t       u 

Both  of  these  results  are  rational ;  therefore  (2)  is  the  rational- 
izing factor. 

27.  Examples. 


I.     Rationalize  d'^- 


With  a  common  denomininator  for  the  exponents  this  becomes 

4       .3  11 

d^  —  f^';  whence  ;/  =  6,  5=4,  /=3  ;  then  x=d^ ,  y=r^ . 

.  ^^-yj(^.r-^ -^x4j/+.r3_j'2 -f  x2j/-^  +.ry'* -f^^; 

('4  11    I       2-0  _1  ri     3  1_:2     fi.  8     9  4     12  \V\ 


f      -2  X  I    f      J  0_  8.    1,  0.    '2_  4     ;',  2     4 

=  \d-'-r'\  [d'^~ +d^r'  ^d""?^  +  d^r^  +  d^r^  ^7^ 

=id^  —  r'^,  which  is  rational. 

2.     Rationalize  6  +  3  v^5. 

With  a  common  denominator  for  the  fractional  exponents  this 
.4  1^  } 

becomes  6"^ +  (3*  X5)^;  whence  w=4,  ^=4,  ^=1  :  then  .v=6'*  and 

1 
1/=  (34  X  5)"^.     Therefore 

(x^y)(x^-x''y^xy''--y^  ) 


48  '  Algebra. 

=  [6*+(3'*X5)*J  f6-'^-6^X3X5^+6X3'^X5^-3-^X5*J 


3- 


=  6*-3*X5=89i 
Rationalize  \^  2  +  2^^9. 


( rta  irr . 


28.  Rationalization  of  the  Denominators  of  Frac- 
TiONS.  The  most  common  application  of  rationalizing  factors  is 
in  the  rationalization  of  the  denominators  of  irrational  fractions. 
Considerable  labor  is  saved  in  computing  the  value  of  a  numeri- 
cal  irrational   fraction  if  we    first   rationalize   the    denominator. 

Thus,  to  compute  the  value  of  — _  -     -_i  correct  to  five  decimal 

places,  three  square  roots  must  be  taken  and  one  of  them  must  be 
divided  by  the  difference  of  the  other  two.  Now,  it  will  be  obvious 
on  reflection  that  these  square  roots  must  be  taken  to  nearly  ten 
places  of  decimals  if  we  are  to  be  absolutely  certain  that  five  deci- 
mal places  of  the  quotient  are  correct.  It  will  be  easily  seen  how 
much  more  readily  the  value  can  be  found  after  the  denominator 
has  been  rationalized.  Multiplying  both  numerator  and  denomi- 
nator by  the  rationalizing  factor  for  the  denominator,  wt  have 

>^5  V  z^{-y  'j-\-\^  2)  v^35-fv/io 

v^7-x/^~"(>/7-v/2)  {'s/~^-\-V~2)~  5 

Now  but  two  square  roots  need  be  taken,  and  these  to  no  more 
than  five  decimal  places,  since  the  exact  value  of  the  denominator 
is  known. 

29.  Examples. 

1.  Rationalize  the  denominator  of  — ::= -. 

v/3-f-\/2 

2.  Rationalize  the  denominator  of    '  —  - — ;=. 

1 

3.  Prove  I :_     =(2  — V^2)"2. 

[2  +  V2 
4..     Given  v^ 3=  1.7320508,  find  the  value  of —. 


Rationalize  the  denominator  of 
Rationalize  the  denominator  of 


Radicals.  49 

v/,8 

cS'.     Rationalize  the  denominator  of  — 

3  +  2n/2 

p.     What  relation  must  hold  between  a  and  x  in  order  that 

a  -{-  \-  X 

s/  2  —  '^'^~}-'^S 
10.     Rationalize  the  denominator  of  -  _—  -  _ -^. 

V  2-HN/3-f>/5 
//.     Compute  the  value   of  the  following  to  three  places  of 
decimals,  having  first  reduced  it  to  its  simplest  form  : 


^34-v^S+^5-x/ 


\^x'-\-i-\-\^x''—i      '^sc'^i  —  s/x'—i 

12.      Prove      ^    -u_-^_        ^_:"_-;_-{-      --^r     =2Ji:^. 

V  x=+  I  —  ^x"—  I      s/ x'-\- 1  -j-  Vx"—  I 

30.  Theorem.   If  a,  b,p,  q  are  commensurable  a7id  >/ b  and^q 
incommensurable,  and  (f  a-\-\^ b  =p-{-\^q  ,  then  a=pands^ b  =v^^. 

\i a  does  not  equal/,  suppose  a=p  +  d.     Substitute  this  value 
for  a  in  the  given  equation,  and  we  have 

p+d+\^b=p-]-\^q 
or  d-{-  \^b  —  >/q 


squaring  both  sides 


whence 


d-  +  2ds/b-\-b=q 
2d 


That  is,  an  incommensurable  quantity  equals  a  commensurable, 
which  is  absurd.  Therefore  a  cannot  differ  from  p.  And  if-fl'=/>, 
\^b  must  equal  ^ q  . 

A— 6 


50  Al^GKBRA. 

31.  ExAMPLKS.     We  append  a  few  miscellaneous  examples  on 
the  last  two  chapters. 

JL        _L        i 
/.     Does  (a-\-x)''=a''-j-x''  f 

2.     Multiply  together  ^a ,  y^\  ^r,  ^ yi^  and  a~'^. 

2  2 

J.     Simplify        -3— -^r  X  1     -  I 

y-\-x" 

4.  Multiply  together 

^x'^^+x"y^y'\  s^J(f—y\  s/x^'^—x^y  +V"  and  \^x''-\-y\ 

5.  Multiply  together 

1  1^  1  i_ 

(a'-^ab+tf-')",   (a—b)",   (a—b)'-   and  (d'-^ab  +  b')- 

(f — x:^ 


a — X 


6.     Simplify     TT^ 


s/a- 


"+i 


7,      Simphfy      y^:^.:..      2  ~+i 

X—  ^  x'—y        y 


^.     Cube  the  expression  a"v/x—%/<^av^jj', 

9.  Prove  2  +  ^3  is  the  reciprocal  of  2— "v/3  ;  and  find  what 
must  be  the  relation  between  the  two  terms  .r  and  x^y  so  that 
jtr-fx/  y  shall  be  the  reciprocal  oi x—^ y. 


10.     Simplyfy     !  — 


.  .   (i-\-xy^(\-xf 

11.  Simphfy  the  expression  ^        1 

(iJrxf-(\-xf 
first  by  rationalizing  the   numerator,   and   then  by  rationalizing 
the  denominator. 

12.  Prove  that  if  p=i  and  ^=5, 

Pq-^e/'-'^-2-\-qp-^e'^-'       3  +  5^'' 


CHAPTER   IV. 

QUADRATIC    EQUATIONS   CONTAINING   ONE   UNKNOWN   QUANTITY. 

1.  Definition.  An  Equation  of  the  Second  Degree,  or  a 
Quadratic  Equation,  is  one  where  the  highest  degree  of  any  term 
with  reference  to  the  unknown  quantities  is  two. 

It  must  be  remembered  that  the  degree  of  an  equation  with 
reference  to  any  quantity  is  not  spoken  of  unlCvSS  the  equation  is 
rational  and  integral  with  reference  to  that  quantity.  See  I, 
Art.  6. 

2.  We  will  consider  in  this  chapter  quadratic  equations  con- 
taining but  one  unknown  quantity,  such  as — 

^  —  3^+5^=24,  (i) 

2x^— |jr=.346,  (2) 

^3  ^^-^-  (f-  ^^l\x=^-  v^  5,  (3) 

m-\-^-\  X'  +  {d—  t)x=p-^  \^k.  (4) 

These  equations  are  all  obviously  quadratics.  But  some  equa- 
tions, which  are  irrational  or  fractional  with  reference  to  x  in 
their  present  form,  drop  into  the  quadratic  type  as  soon  as  the 
proper  transformations  are  performed.     Thus  the  equation, 

y/ 1,—  ^  a        I     .  V.a—>/b 
^x+ =        -f    -       ..  (5,  a) 

\^  b  ^x  \/ a 

may  be  made  integral  with  reference  to  x  by  multiplying  through 
by  s/ X,  the  resulting  form  being 

(y/ fj-y/aWx         ,  Wa-s/ bWjt 

"+- ^1 = '  ^  —^--~  ' 

Transposing  and  uniting  terms, 

\^ab 
Transposing  the  rational  parts  to  the  right  hand  side  of  the  equa- 
tion, we  obtain  the  form 

{b—a)^x_    _ 
y^al      ~' 


52  Algebra. 

Now  rationalizing  with  respect  to  x,  by  squaring  both  sides  of 
the  equation,  it  becomes 

at? 
Finally  transposing  and  collecting  terms,  we  have 

,  a'  +  ^' 

which  is  a  quadratic  equation.  While  the  equation  (5,  a)  has 
been  reduced  to  the  quadratic  form  (5,  d)  by  apparently  legitimate 
processes,  yet  we  will  find  that  the  integralization  and  rationaliza- 
tion" of  an  equation  with  reference  to  the  unknowai  quan- 
tity has  in  general  an  effect  on  the  solution  of  the  equation 
which  it  is  necessary  to  take  into  account,  and  which  renders  it  pos- 
sible that  the  values  of -r  which  satisfy  (5,  b)  may  not  be  identical 
with  those  that  satisfj^  (5,  a).  For  this  reason  the  treatment  of 
those  equations  which  require  the  operation  of  integralization  or 
of  rationalization  before  they  are  in  the  quadratic  form,  is  reserved 
for  Chapter  VI. 

3.  Typical  Forms  ok  the  Quadratic.  It  is  evident  that 
equations  (i),  (2),  (t,),  (4.)  and  (^5,  d),  or  any  other  quadratic 
equations  which  can  be  imagined,  may  all  be  said  to  be  of  the 
typical  form, 

ax''-\-dx=c,  (6) 

where  a,  d  and  c  are  supposed  to  stand  for  any  numbers  whatever, 
either  integral  or  fractional,  positive  or  negative,  or  commensur- 
able or  incommensurable.  Hence  ax'-[-dx=^c  is  said  to  be  a 
typical  form  of  the  quadratic  equation. 

If  we  suppose  the  quadratic  equation  to  be  divided  through  by 
the  coefficient  of  x:^  the  result  will  be  of  tlie  form 

x'-\-px=q,  (7) 

where  /  and  q  are  supposed  to  be  any  algebraic  quantities  what- 
ever, fractional  or  integral,  positive  or  negative,  commensurable 
or  incommensurable.  This  is  the  second  typical  form  of  the 
quadratic  equation,  and  one  which  is  much  used. 

4.  Definition.  A  /vV(?/of  an  equation  is  any  value  of  the  un- 
known quantity  which  satisfies  the  equation. 

Thus  \  isa  root  of  the  equation  3;^— 6=0,  for  when  substituted 


Quadratic  Equations.  5,-^ 

for  X  it  satisfies  the  equation.  Also,  both  2  and  3  are  roots  of  the 
equation  .r^— 5.r+ 10=4,  for  either  of  these  values  when  substitu- 
ted for  X  will  satisfy  the  equation. 

The  student  must  carefully  note  that  this  is  an  entirely  different 
use  of  the  word  7vot  from  that  occuring  in  the  expres.sions  square 
root,  cube  root,  etc. 

5,  Equations  of  the  second  degree  are  often  divided  into  the 
two  classes  of  complete  and  mcoynplete  quadratics.  A  complete 
quadratic  is  one  which  contains  both  the  first  and  second  powers  of 
the  unknown,  as  x~-\-px^q.  An  incomplete  quadratic  has  the 
first  power  of  the  unknown  quantity  lacking,  and  hence  can  al- 
ways be  placed  in  the  fonn  x"—q,  where  q  is  any  algebraic  quan- 
tity conceivable.  By  some  the  adjectives  affected  and  pure  are 
used  in  place  of  the  words  complete  and  ijicomplete  respectively. 

6.  Problem.     To  Solve  anv  Incomplete  Quadratic. 
First,  reduce  to  the  form 

x'^=q 
by  putting  all  the  known  quantities  on  the  right  hand  side  of  the 
equation  and  all  the  terms  containing  x'^  on   the  left  hand  side, 
then  dividing  through  by  the  coefficient  of  x\ 

Then  take  the  square  root  of  both  sides  of  the  equation,  remem- 
bering that  every  quantity  has  two  square  roots,  and  we  obtain 

x=±^q 
and  the  equation  is  solved. 

It  might  be  thought  that  in  taking  the  square  root  of  both  sides 
of  x'^q  we  should  write 

But,  by  taking  the  signs  in  all  possible  ways,  this  givesi 

j^x=  +  s/q 
—x=^  —  ^q 
Jrx^-s/q 
—x=-\->/q. 


54  Algebra. 

Each  of  the  first  two  of  these  is  equivalent  to  x=  ^ q  ,  and  each 
of  the  last  two  is  the  same  as  x=  —  \^q,  and,  on  the  w^hole,  we 
merely  have 

Whence  it  is  seen  to  be  sufficient  to  write  the  sign   ±  on  but 
one  side  of  the  equation. 

7 .  Examples  of  Incomplete  Quadratics. 
Solve  the  following  equations  : 

2.     (ax~.b)(ax-\-b)=c. 

J.        (x+2)^-h(x-2r=24. 

/.  (x-\-d/-j-(x-dr=r. 

5.  (ax-\-d)^-{-(ax — bf=^c. 

6.  (x+T)(x-g)  +  (x-'j)(x-^^)=^'j6. 

7.  (x-{-a)(x-b)  +  (x-a)(x-^b)=c. 

8.  (x-^a)^=q. 

Show  that  examples  i ,  3  and  6  may  be  solved  by  proper  substi- 
tution in  the  results  to  examples  2,  5  and  7  respectively. 

8.  We  have  solved  the  equation  x"=^q,  and  also  the  equation 
(x-{-a)^—q  (Ex.  8)  in  a  similar  manner.  Now  it  is  evident  that 
the  equation 

x'-\-px=q 
can  be  solved  if  it  can  be  put  in  either  of  the  above  forms.  It  can 
be  placed  in  the  form  (x-\-a)^=q  if  the  first  member  can  be  made 
the  square  of  a  binomial.  On  inspection  it  is  seen  that  x"-\-px  are 
the  first  two  terms  of  the  square  of  a  binomial,  the  third  term  of 
which  must  be  ^/>^  Hence,  if  we  add  \p'^  to  both  sides  of  the 
equation 

x^J^px=q. 
it  takes  the  form 

x'+px+\p^=q+\p\ 

or  ^  r-^+i/>/=^+i/^ 

which  is  of  the  form  (x-\-a)-=q. 

The  process  of  putting  a  quadratic  equation  in  this  form  is 
called  completi7ig  the  square. 


Quadratic  Equations.  55 

9.  Probi^km.  To  Solve  thk  Typical  Quadratic  x'-{-px—q. 
Add  ^/>^  to  both  members  and  we  obtain 

The  left  hand  member  is  seen  on  inspection  to  be  the  square  of 
the  binomial  (x-\-yP)  ;  whence  taking  the  square  root  of  both 
members, 

Solving  this  simple  equation  for  x  we  have 

x=-\P^^q+\p% 
w4iich  gives  the  two  values  of  .1 , 

-\P+^(1-^\P'  and  -^p-Vq-^\p\ 
Hence,  to  solve  an  equation  in  the  form  j^-\-px=-q,  add  the  square 
of  07ie-half  the  coef[icie7it  of  x  to  each  side  of  the  equation.      Take  the 
square  root  of  both  me7nbers,  and  an  equatio?i  of  the  first  decree  is 
obtained,  from  jvhich  x  can  be  found  in  the  usual  way. 

10.  Problp:m.  To  vSolve  thk  Typical  Quadratic  ax^-\-bx 

Multiply  through  by  4<3!  and  obtain 

^cC\x^  4-  \abx^  ^^ac. 
Adding  b^  to  both  members  it  becomes 

\a^x^  +  d^a  bx  -f-  b- = ^ac-\-  b^ . 
The  left  hand  member  is  seen  on  inspection  to  be  the  square  of  a 
binomial  ;  whence,  taking  the  square  root  of  both  members,  we 
obtain 

2ax  -{-  b—±^  /i,ac-\-  b\ 
whence,  solving  this  simple  equation, 

—  b^sf'^c'^lf 
2a 
which  gives  the  two  values  of  .v, 

—  ^-f  v^4«r+A-'        ,  —b—y/Atac-\-b' 

and  

2a  2a 

Hence,    to  solve  an   equation   in   the  form   ax'-\-bx==c,  multiply 

through  by  four  times  the  coefficient  of  .v-  and  add  the  square  of  the 

coefficient  of  x  to  each  side  of  the  equation.      Then  take  the  square 

root  of  both  members,  aiid  an  equation  of  the  first  degree  will  be  ob- 

taiyied,  from  ivhich  x  can  be  found. 


56 


Algebra. 


i 


Ife 


»b^ 


11.  HiSTOiiiCAii  Note.  The  origin  of  the  solution  of  the  quadratic  equa- 
tion cannot  be  definitely  traced  to  any  one  man  or  any  one  race.  Algebra,  as 
we  now  have  it,  has  been  a  slow  growth,  and,  as  we  pass  ba(jk  in  time,  it  grad- 
ually shades  off  into  the  arithmetic  of  antiquity.  Diophantus,  an  Alexandrian 
Greek  of  the  fourth  century,  A,  D.,  who  wn  te  a  treatise  cm  arithmetic,  could 
undoubtedly  solve  <piadratic  equations,  although  he  devotes  no  .special  book 

their  treatment.  But  algebra,  in  a  more  perfect  form,  may  be  traced  to 
the  Hindoos.  Aryabhata  (475  A.  D.)  was  familiar  with  a  solution  of  the  com- 
plete quadratic,  and  Bramagupta  (59S  A.  D.)  gives  a  comparatively  elaborate 
treatment  of  it.  The  solution  ofthe  quadratic  was  also  known  to  the  Arabs,  and 
a  solution  with  geometric  treatment  is  given  by  Mohammed  ben  Musa,  of  the 
ninth  century.  All  the  early  methods  of  solution  consist  in  what  is  commonly 
known  as  "conM)leting  the  square"  and  wore  substantially  the  same  as  those 

12.  Examples  of  Complete  Quadratic  Equations,  If  a 
quadratic  equation  cannot  be  placed  in  the  fovu]  x' -\-px=q  icithotit 
the  int7-oductio7i  of  fractions,  it  is  generally  advisable  before  solu- 
tion to  clear  it  of  fractions  thereby  putting  it  in  the  form  ax'-\-bx 
=  r,  in  which  case  a,  b,  and  c  will  be  integral.  The  equation  can 
then  be  solved  by  the  method  of  Art.  lo,  thereby  avoiding  frac- 
tions in  the  process  of  solution,  which  is  a  great  advantage.  If 
the  equation  takes  the  form  x^--\-px=^q  without  /;  and  q  being 
fractional,  then  a  solution  by  the  method  of  Art.  9  will  be  better. 

To  illustrate  the  common  arrangement  of  the  work  we  solve  the 
following  quadratic. 
Find  the  valties  of  x  in 

10— A^^ 

3 
Clearing  of  fractions, 

gx^—2'jx-\-i^=  10— X". 
Transposing  and  uniting  terms. 

Multiplying  through  by  4  times   10,  and  ad- 
ding (2'jy  to  both  sides,  we  obtain 

400X''—()x-{-(2'j/=  —  200-\-(2'j)\ 

or  4oo.v"—('>- 4- 27^=529. 

Taking  the  square  root  of  both  members, 

20-X"— 27=^1=23, 
and  solving  this  simple  equation 


The  first  task  is  to 
place  the  eqiiation  in 
one  of  the  typical 
forms. 


Now  "complete  the 
square." 


■9-'i+5  = 


Since  it  hi  been 
proved  that  thi  meth- 
od will  give  a  complete 
square,  it  is  not  nec- 
essary to  work  out 
the  value  of  the  co- 
efficient of  X,  butmere- 
ly  to  indicate  it  by  (). 


2o.x-=5o  or  4 
x=  2^-  or  i. 


Quadratic  Equations.  57 

Solve  the  following : 
/.     .v-^  4-7.^4-15  =  5. 
2.     X-  4-  6x—  I  =  5  —  20A-. 
3'     3^^"+5-v==ioo. 
4..     1 5^-"— 28.^4-10=5. 
In  completing  the  square  in  a  case  like  this  where  the  coeffi- 
cient  of  jf   is  divisible  by  2,  fractions  can  be   avoided  without 
multipljang  the  equation  through  by  4.     Thus: 

15^^— 28x=  — 5, 
multiplying  through  by    15  (instead  of  4X15),  and  adding   the 
square  of  one-half  of  28,  we  obtain 

(i5r-i:=- 15x28-1-4- (14)'= -5x15  + (14)' 
w^hich  is  a  complete  square. 

5.     A-+i2yV^'=38|. 

7.  x^4-6.5i  =  5.2x. 

8.  x^--\-\—ax—-=o. 

9'  (■'^—?>)(^^—S)  =  o. 

10.  (x—a)(x—d)=o. 

11.  (zx—^)(^x—2,)=o. 

12.  (ax—b)(bx—a)=o. 

ij.      (x-\-a  —  b)(x—a-\-b)  =  o. 
14..     a'—x^=^(  a—x  )(  b-\-c—x  ) . 

/5-     r33+io.r/+r56+io-t-/=r65+i4^C. 

16.     r7-4v/3;.r=4-r2-\^3A=?- 

18.  x^-\-qax=a'' — b'. 

ip.  (x—a/=(x—b)(a  +  b). 

20.  dez'—(d'-^e')2^-de=o, 

21.  x''—2a(x-\-b)=^2bx—a'—b'. 

22.  .r'4-ioji--f  30=5. 

2j.  a-^b-\-x— a'b'x'. 

x^  x"" 

24..  -\-ax='    +bx. 

^        a  b 


25.     ax'-\-bx-\rC=x'-^px-\-q. 


K—l 


\ 


\ 


58  Algebra. 

26 .     x~  —  I  =  k(kx' — ^x—k) . 

2y.     (x—a)(x—b)-\-(x—b)(x—c)-\-(x—c)(x—a)=o. 
Result,  x=\(a  +  b+c)±iya'  +  b''  +  r-—ab—bc—ca. 
28.     ji-'^  +  6x4-2i  =  io. 
This  becomes,  in  the  typical  form, 

x^  +  6;i"=  — II. 
Completing  the  square,  we  obtain 

x^-^6x-\-g—  —  2. 
Now  we  cannot  obtain  the  square  root  of  the  right-hand  mem- 
ber of  this  equation ;  for  it  is  a  negative  quantity,  and  the  square 
of  no  algebraic  number  can  be  negative.  But,  if  we  were  to  go 
through  the  operation  of  finding  x  as  has  been  done  in  the  other 
cases  above,  and  indicate  the  root  of  —2  as  if  we  coicld  take  it, 
we  would  have 

x=  — 3±v/— 2. 
Thus  we  have  had  forced  upon  us  in  the  solution  of  the  quad- 
ratic equation,  something  which,  whatever  interpretation  it  may 
have,  is  evidently  7iot  a?i  algebraic  quantity  in  the  sense  in  which 
the  term  is  commonly  used.  Such  an  expresion  is  called  an  im- 
aginary, and  its  treatment  is  reserved  for  Part  II  of  the  pres- 
ent work.  In  the  next  chapter  will  be  found  a  discussion  of  the 
circumstances  under  which  such  expressions  occur. 
2g.     4Jt:^  +  4-r+4=jr^. 

13.  Problems  Requiring  the  Solution  of  Quadratic 
Equations.  The  student  in  his  previous  study  has  probably 
already  noticed  that  the  first  task  in  the  algebraic  solution  of  a 
problem  is  always  an  attempt  to  express  the  language  of  the  prob- 
lem in  algebraic  symbols  ;  that  is,  to  cast  the  relations  and  condi- 
tions expressed  by  the  words  of  the  problem  into  an  equivalent 
statement  in  the  form  of  one  or  more  algebraic  equations.  This 
w^ork  is  called  the  statement  of  the  problem,  and  is  generally  a 
difficult  one  for  the  beginner  to  perform.  When  the  statement  of 
a  problem  is  complete,  all  that  remains  to  be  done  is  the  solution 
of  the  equation  or  equations  obtained  thereby  by  processes  already 
familiar. 

We  wish  to  strongly  emphasize  the  fact  that  the  equation 
obtained  by  the  translation  of  the  words  of  most  of  the  algebraic 
problems  in  the  books  is  often  7iot  an  exact  equivale?it  to  the  condi- 


Quadratic  Equations. 


59 


tions  and  relatio7is  told  in  the  language  of  the  problem.  In  fact, 
the  equation  often  eynbraces  more  than  the  problem  itself.  We 
will  illustrate  this  by  the  following  problem  : 

A  certain  number  consists  of  two  digits  whose  sum  is  lo.  If 
we  reverse  the  digits  and  multiply  this  new  number  by  the 
original  number,  the  product  will  be  2944.  Required  the  number. 
I^et  -r=the  digit  in  unit's  place  ; 
then  10— -v=the  digit  in  ten's  place, 
and  10(^10— Aj  =  the  value  of  the  digit  in  ten's 
place  ; 

whence  10(^10— ;rj-{-;r= the  value  of  the  orig- 
inal number ; 

also  lox+i^io— -rj=the  value  of  the  number 
with  the  digits  reversed. 
I       But,  by  the  problem, 

I       [lorio— -i-;-f.r]  [loi-f  ("lo— -^vj]=2944.   (\) 
\  That  is. 


Statement,  or  trans- 
lation of  the  language 
into  an  algebraic  equa- 
tion. 


Solution  of  the  cqim 
tion. 


(^  100— 9JiJ(^  lo-j-  9.rj=  2944. 
Expanding  left  member, 

8 IX''— 8  lox— 1000=  —  2944. 
Transposing  and  uniting, 

8iA^— 8iox=  — 1944. 
Dividing  through  by  81, 

x^— iox=  — 24.  (2) 

Completing  square  and  solving, 
-t==4  or  6. 

The  number  is  therefore  either  46  or  64.  Now  consider  equa- 
tion (i)  as  a  translation  of  the  problem  into  algebra.  As  far  as 
is  stated  by  the  eqiiation  ( i )  the  unknown  quantity  x  may  be  an}' 
algebraic  quantity  conceivable, — positive  or  negative,  integral  or 
fractional,  rational  or  irrational,  or,  in  fact,  it  may  possibly  be 
what  we  have  called  an  imaginary.  As  far  as  the  equation  ex- 
presses the  nature  oi  x,  it  may  as  likely  turn  out  in  the  solution 
one  kind  as  another  of  those  enumerated.  But,  as  expressed  in 
the  language  of  the  problem,  x  must  be  a  digit ;  that  is,  a  positive 
integral  7iumber  less  thaji  ten.  The  equation  does  not  express 
this  fact  and  cannot  be  made  to  do  it.     The  reason  why  the  prob- 


6o  Algebra. 

lem  really  works  out  all  right  is  that  it  was  made  to  order ;  that  is, 
the  number  2944  was  especially  selected  so  that  the  problem 
would  * '  work  out  " .  If  we  wish  this  problem  stated  in  words  so 
that  it  is  more  nearly  identical  with  its  expression  in  the  form  of 
an  equation,  we  must  throw  out  the  word  "digits"  as  follows: 

There  are  two  numbers  whose  sum  is  ten.  If  ten  times  the 
first  plus  the  second  is  multiplied  by  ten  times  the  second  plus 
the  first,  the  product  will  be  2944.     Find  the  numbers. 

This  is  nearly  as  general  as  the  algebraic  equation.  It  permits 
of  either  positive  or  negative,  integral  or  fractional,  commensur- 
able or  incommensurable,  results,  and  indeed  as  the  word  num- 
ber is  often  used  it  would  permit  of  imaginary  results.  This  prob- 
lem can  be  made  identical  with  the  original  by  adding  at  its  close 
some  such  caution  as  this  : 

Do  not  obtain  a  fractional,  a  negative,  nor  an  incommensur- 
able result,  nor  any  result  greater  than  9. 

It  is  such  conditions  as  these  that  we  fail  to  incorporate  into  an 
algebraic  equation.  The  algebraic  statement,  as  far  as  the  un- 
known is  concerned,  is  always  the  most  general  possible  and  con- 
tains in  it  no  restriction  of  the  unknown  to  any  particular  class  of 
numbers,  and  for  this  reason  the  algebraic  statement  of  a  problein  is 
often  more  general  thaii  the  problem  itself.  This  fact  should  be  re- 
membered, as  it  will  help  to  explain  many  apparent  difficulties 
which  arise  in  some  problems.  These  non-algebraic  conditions  in 
a  problem  must  be  ignored  until  after  the  solution  is  had,  and 
then  if  a  result  is  obtained  like  a  fractional  number  of  live  sheep 
or  a  negative  price  per  head,  it  must  be  cast  out,  not  because  the 
mathematics  is  unreliable,  but  because  the  problem  is  cramped 
and  does  not  fill  up  the  full  measure  of  generality  which  algebraic 
methods  provide  for. 

The  greatest  breadth  and  elegance  of  algebraic  analysis  would 
be  observed  in  the  treatment  of  problems  in  geometry,  mechanics 
and  physics,  but  since  we  cannot  presume  any  considerable  famil- 
iarity with  these,  only  problems  involving  the  simplest  geomet- 
rical principles  have  been  inserted.  While  the  elegance  of  alge- 
braic methods  is  best  seen  in  the  solution  and  discussion  of 
problems  of  equal  generality  with  their  algebraic  statement,  yet 
those  we  give  are  not  entirely  of  this  class. 


Quadratic  Equations.  6i 

PROBLEMS. 

/.  The  hypothenuse  of  a  right  angled  triangle  is  io,and  the 
excess  of  the  perpendicular  over  the  base  is  2.  Find  the  sides  of 
the  triangle. 

2.  The  hypothenuse  of  a  right  angled  triangle  is  h,  and  the 
excess  of  the  perpendicular  over  the  base  is  c.  Find  the  sides  of 
the  triangle. 

Can  e  in  this  problem  be  assigned  a^iy  value  whatevei  ? 

J.  The  perimeter  of  a  rectangle  is  16  feet,  and  its  area  is  15 
square  feet.     Find  the  dimensions  of  the  rectangle. 

4.  The  perimeter  of  a  rectangle  is  p  feet,  and  its  area  is  a 
square  feet.     Find  the  dimensions  of  the  rectangle. 

Show,  from  the  result,  that  the  square  is  the  greatest  possible 
rectangle  which  can  be  made  with  a  given  perimeter. 

5.  The  sum  of  the  squares  of  three  consecutive  odd  numbers 
is  83.     Find  the  numbers. 

What  would  you  say  in  case  the  number  56  was  given  in 
place  of  83? 

Make  the  problem  read  so  that  56  will  be  allowable. 

6.  The  sum  of  the  squares  of  four  consecutive  even  num- 
bers is  120.     Find  the  numbers. 

7.  If  962  men  were  drawn  up  in  two  squares,  and  it  were 
found  that  one  square  had  18  more  ranks  than  the  other,  what 
would  be  the  size  of  each  square  ? 

8.  A  boat's  crew  row  3^  miles  down  a  river  and  back  again 
in  I  hour  and  40  minutes.  Supposing  the  river  to  have  a  cur- 
rent of  2  miles  per  hour,  find  the  rate  at  which  the  crew  would 
row  in  still  water. 

What  do  you  say  about  tlie  negative  result  ? 

p.  A  boat's  crew  row  d  miles  down  a  river  and  back  again 
in  /  hours.  Supposing  the  river  to  have  a  current  of  r  miles  per 
hour,  find  the  rate  of  rowing  in  still  water. 

Show  from  the  result  that  the  problem  will  always  give  one 
positive  and  one  negative  value  of  x  for  all  values  of  d,  t  or  r. 


62  Algebra. 

lo.  The  total  area  of  two  squares  is  a  square  feet.  A  side 
of  one  square  is  found  to  differ  from  a  side  of  the  other  by  d  feet. 
Find  the  side  of  each  square. 

Is  this  problem  possible  for  all  values  of  (T^? 

//.  Two  trains  are  dispatched  from  a  station,  one  starting  an 
hour  before  the  other.  The  rate  of  motion  of  the  later  train  is  5 
miles  per  hour  more  than  that  of  the  other,  and  it  overtakes  the 
first  train  at  a  distance  of  150  miles  from  the  station.  Find  the 
rate  of  motion  of  each  train. 

12.  Generalize  the  foregoing  problem  and  solve  it.  DiscUvSS 
the  results. 

13.  A  rectangular  metal  plate  is  20  inches  longer  than  wide. 
It  is  expanded  by  heat  until  each  dimension  increases  by  2V  ^^  ^^s 
former  length,  thereby  increasing  the  area  of  the  plate  246  square 
inches.     Find  the  original  dimensions  of  the  plate. 

14..  A  man,  bom  in  1806,  died  at  the  age  of  x  in  the  year 
x^     When  did  he  die  ? 

75.  Two  trains  pass  at  a  junction.  One  is  traveling  south  at 
the  rate  of  30  miles  an  hour  and  the  other  is  traveling  west  at  the 
rate  of  40  miles  per  hour.  How  long  before  the  two  trains  are 
100  miles  apart? 

Interpret  the  two  results. 
16.  Two  trains,  A  and  B,  are  traveling  on  roads  at  right 
angles  to  each  other,  each  approaching  the  crossing.  A  is  10 
miles  from  the  crossing  and  traveling  uniformly  30  miles  an 
hour,  while  at  the  same  instant  B  is  20  miles  from  the  crossing 
and  traveling  uniformly  40  miles  an  hour.  When  will  they  be  5 
miles  apart  ? 

Explain  the  two  results. 
ly.  Two  trains,  A  and  B,['Sir^  traveling  on  roads  at  right 
angles  to  each  other.  A  is  J40  miles  from  the  crossing  and 
is  moving  towards  it  at  the  ||uniform  rate  of  30  miles  an  hour. 
B  is  20  miles  from  the  crossing  and  is  moving  frofji  it  at  the  uni- 
form rate  of  25  miles  an  hour.  At  what  times  are  the  trains  90 
miles  apart  ? 

Interpret  the  results. 


Quadratic  Equations.  63 

18.  Along  the  sides  of  a  right  angle  two  bodies,  A  and  /y, 
move  with  unifonii  velocity.  A  \^  a  miles  from  the  vertex  and 
moving  p  miles  per  hour,  while  at  the  same  instant  B  '\^  b  miles 
from  the  vertex  and  moving  q  miles  per  hour.  At  what  times  are 
the  two  bodies  d  miles  apart  ? 

Show  that  the  result  obtained  can  be  used  as  a  formula  to 
solve  Prob.  16. 

Show  that  by  giving  the  proper  interpretation  to  q,  as  to  its 
positive  or  negative  character,  that  the  formula  can  be  made  to 
solve  either  Prob.  16  or  17  at  will. 

Under  what  conditions  will  the  bodies  7iev€r  be  d  miles  apart  ? 

7^.  Two  circles,  A  and  B,  move  wdth  their  centers  always 
on  the  sides  of  a  right  angle.  A,  whose  radius  is  -R  feet,  is  a 
feet  from  the  vertex  and  moving  uniformly  p  feet  per  second.  B, 
whose  radius  is  rfeet,  is  b  feet  from  the  vertex  and  moving  uni- 
formly q  feet  per  second.  At  w^hat  times  are  the  circles  tangent 
to  each  other? 

Result :  Tangent  externally  in 

ap^bq^s/(RJrrnp^±^)^-(apJjqr  ^^^^.^ 
p^-^q^       ~  seconas. 

Tangent  internally  in 


ap^i;q±,V(/^-rr(p^  +  q^)  +  (ap-^bqr-  ^^^^^^^^ 

P"  +  f 
Show  that  it  is  possible  for  them  to  be  tangent  externally  and 

not  tangent  internally. 

Show  that  it  is  impossible  for  the  circles  to  be  tangent  inter- 
nally without  first  being  tangent  externally. 

Show  that  the  known  quantities,  may  have  such  values  that 
the  two  circles  will  never  be  tangent  at  all. 

20.     Find  the  side  of  an  equilateral  triangle,  knowing  that  a 
side  exceeds  the  altitude  by  d  feet. 


CHAPTER  V. 

THEORY  OF  QUADRATIC  EQUATIONS  AND  QUADRATIC  FUNCTIONS. 

I.  It  follows  immediately  from  the  definition  (I  Art.  4  )  that 
every  rational  integral  quadradic  function  of  x  is  of  the  form 

/x^-\-72X-\-r 
where  f,  n  and  r  stand  for  any  algebraic  numbers  whatever,  posi- 
tive   or     negative,     integral    or    fractional,     commensurable    or 
incommensurable. 

If  we  take  the  typical  quadratic  equation 

ax''-\-bx=^c 
and  transpose  the  c  to  the  left-hand  side  of  the  equation  it  becomes 

ax'^  -\-  bx — r=  o . 
This  can  obviously  be  said  to  be  of  the  form 

lx^-\-nx-\-r=^o 
and  consequently  a  quadratic  equation  may  be  defined  as  an  equa- 
tion which  can  be  placed  in  the  form  of  a  rational  integral  quad- 
ratic function  equal  to  zero. 

Since  a  root  of  an  equation  has  been  defined  as  any  expression 
which  substituted  for  the  unknown  will  satisfy  the  equation,  there- 
fore it  is  evident  from  the  form 

ax^-\-bx — <:=o 
that  a  root  of  a  quadratic  equation  may  also  be  stated  to  be  an 
expression  which  substituted  for  x  causes  ax--\-bx—c  to  equal 
zero ;  that  is,  causes  the  function*  oi  x  to  vanish. 

Hence  we  may  say:  A  quadratic  equatioyi  is  a7iy  equation  ivJiich    _^ — ^ 
can  be  put  in  the  form  of  a  rational  ititegral  quadratic  fu7ictio7i'7quaT  \ffi^ 
to  ze7'-o,  a7id  a  root  of  it  is  a7iy  expression  ivhich,  substituted  for  x, 
causes  the fu7ictio7i  of  x  to  vanish. 

Thus  the  equation  ..r^— 3Jt:=io,  whose  roots  are  5  and  —2,  when 
placed  in  the  form  of  a  function  of  x  equal  to  zero,  becomes 

x'^ — 3X — 10=0. 
It  is  now  seen  that  the  roots  are  such  quantities  that,  when  sub- 

*Because  of  the  array  of  adjectives  in  the  expression  '  rational  integral  qnadrati( 
function  of  x  "  we  shall  often,  for  the  remainder  of  this  chapter,  use  the  expression  "func- 
tion of  X  "  in  its  place. 


Theory  of  Quadratics. 


65 


stituted  for  x,  cause  the  function  of  x  to  vanish.     For  the  func- 
tion of  X  is 

X^—'TfX—lO 

and  putting  5  for  x  it  becomes 

25—15—10 
which  is  zero.     Putting  —2  ior  x  the  function  of  x  becomes 

4+6—10 
which  is  also  zero. 

If  anything  else  than  a  root  is  put  for  x  the  function  will  not 
vanish  ;  thus  when 

-r=— 4,  function  of  x  becomes  16+12—10=      18 


x=- 

-3, 

'            "           9+   9—10=        8 

[x=- 

-2, 

'           "           4+   6—10=       0] 

x=- 

-I, 

1+   3—10=—   6 

.r= 

0, 

'           *'           0+   0— io=  — 10 

x= 

I,          "            * 

'           "            I—  3—10=  — 12 

x= 

2, 

'           *'           4—   6— io=  — 12 

x= 

3, 

'           "           9—  9—10=  — 10 

x= 

4- 

'           "          1,6—12—10=—  6 

lx= 

5, 

25—15—10=       0] 

x= 

6, 

36—18—10=        8 

2.  If  we  suppose  the  quadratic  function  divided  through  by 
the  coefficient  of  x''  it  may  be  represented  by 

x^-]-ex-\-f. 
If  we  take  the  quadratic  x^-\-px=q,  and  transpose  the  q  to  the 
other  side  of  the  equation,  we  obtain 

x'^-\-px—q=o 
where  the  left  member  is  seen  to  be  of  the  form  x^-^ex-{-f.     Then, 
since  every  quadratic  may  be  put  in  the  form  x--\-px=q,  it  may 
also  be  placed  in  the  form 

x^-]-px—q=o 
or  better  x^-\-ex-{-f=o. 

In  either  of  the  quadratic  functions 

lx--\-7ix-\-r 
or  xr-\-cx-\-/ 

the  term  which  does  not  contain  x,  that  is  r  or  /,  is  called  the  ab- 
solute term. 

A— 8 


66  AI.GEBRA. 

3.  By  solving  the  equation 
it  will  be  found  that  its  roots  are 


-i^+  V-K-/  and  -}^e-\/\e^-/. 

4.  ThEORKm.     Every  quadratic  fun ctio7i  of  x  can  be  i-esolvcd 
into  the  product  of  tivo  linear  functioiis  of  x. 

Take  the  function  of  Jt:  in  the  form 

x^-\-ex-^f. 
add  and  subtract  \e:'  from  the  function,  thus  not  altering  its  value. 
We  obtain  then  _ 

^  x^-^ex+\e^-\e^+f 

This  may  be  written 

(x+\er-(\e^-f), 
or,  if  we  please,  as  the  difference  of  two  squares, 

{x+w-WV^T- 

Writing  this  as  the  product  of  the  sum  and  difference,  it  takes 
the  form 

or  (x+\e-\^\r-f){x+\e+  Vi^-/), 

which  is  the  product  of  two  linear  functions  of  x. 

5.  Examples.     Resolve  the  following  quadratic  functions  into 
the  product  of  two  linear  functions  of  x  : 

1.  x^— jr— 2IO. 

2.  3.r"+2Ji:— 85. 
J.  x''—6bx-\-gb-. 
4..  /\.a^x^—4ax-\-i. 
5.  .r=-i4.r+33. 

6.  Theorem,     /f  the  roots  of  a  quadratic  equation  are  a  and  b, 
then  the  equation  may  always  be  put  i?i  the  form  (x—a)(x—b)^o. 

By  Art.  4,  the  equation 

x''-\-ex-\-f=o  (i) 

may  always  be  placed  in  the  form 

U-+if  +  \/i^-/)Gr+i^-  Vi?=7)=o.      (2) 


Theory  of  Quadratics.  67 

If  we  represent  the  two  roots  of  (i)  by  «  and  b  for  the  sake  of 
brevity,  we  see  from  Art.  3,  that 

Substituting  these  in  equation  (2),  it  becomes 
(x—a)(x—b)  =  o. 

7.  C0ROI.LARY.  If  all  the  terms  of  a  quadratic  be  transposed  to 
one  side,  that  member  is  exactly  divisible  by  x  minus  a  root. 

8.  Corollary.  The  form  (x—a)(x—b')=^o  may  be  used  iyiter- 
changeably  with  x'' -\- ex -\-f=  o  to  represent  any  quadratic  equation. 

9.  Theorem.  Every  quadi^atic  equation  with  one  unknown 
q2ia7itity  has  two  roots  and  only  two. 

It  has  been  shown  that  every  quadratic  equation  can  be  placed 
in  the  form 

(x—a)(x—b)=o. 

This  equation  is  satisfied  when  the  left  member  is  zero.  But 
the  left  member  becomes  zero  when  either  one  of  its  two  factors 
is  zero  ;  that  is,  when  x=^a  or  x=b.  Because  each  of  these  two 
values  of  x  satisfies  the  equation  it  has  two  roots.  But  the  equa- 
tion can  have  no  other  root ;  for  if  any  other  value  than  a  or  bh^ 
assigned  to  x,  neither  of  the  factors  will  be  zero,  and  consequently 
their  product  will  not  be  zero.  Hence  there  can  be  no  more  than 
two  roots. 

It  is  not  claimed  that  there  must  be  two  different  roots.  In 
fact,  there  is  nothing  in  any  of  the  reasoning  thus  far  which  shows 
that  a  and  b  must  always  have  different  values.  In  general,  they 
are  different  from  each  other,  but  a  special  case  would  be  where 
they  are  alike.     In  this  case  the  quadratic  takes  the  form 

(x—a)(x—a)^o, 
and  we  still  speak  of  two  roots  because  there  are  two  factors  and 
because  it  is  merely  a  special  case  of  the  general  truth.  To  say 
that  an  equation  has  two  roots  equal  to  each  other  is  merely  an- 
other way  of  saying  that  there  is  but  one  value  which  satisfies  the 
equation. 


68  Algebra. 

10.  Theorem.  Whe7i  a  quadratic  eqiiatio7i  is  in  the  form 
x^ -\- ex -\-/=  o ,  the  coefficie?ii  of  x  with  its  sign  changed  equals  the 
sum  of  the  two  roots,. and  the  absolute  term  equals  the  product  of  the 
tzvo  roots. 

The  two  roots  of  the  equation 

x^-\-ex-\-f=o 
are  —\e-\->f\e^—f 

and  —\e—s^\(f  —f 

—  e-\-  o 
Adding  them,  their  sum  is  seen  to  be  — <?,  or  the  coefficient  of  .r 
with  its  sign  changed. 

Multiplying  the  two  roots  together,  recognizing  the  product  of 
a  sum  and  difference,  we  obtain 

which  is  the  absolute  term  of  the  equation. 

Another  Method. 

(a).  First,  suppose  the  two  roots  not  equal  to  each  other. 
Call,  for  abbreviation,  the  two  roots  of  the  equation  x^-\-ex-\-f=o 
a  and  b.     Then,  by  the  definition  of  a  root,  we  have 

a'-\-ea-\-f=o  (i) 

and  b'  +  eb-\-f=o.  (2) 

Subtracting  (2)  from  (i)  we  obtain 

a^-—b'-\-e(a  —  b)  =  o,  (2>) 

or,  dividing  through  by  a—b, 

a-\-b+e=o, 
or  e=  —  (a^b).  (4) 

That  is,  the  coefficient  of  x  is  the  sum  of  the  roots  with  opposite 
signs. 

Now  substitute  this  value  of  <?  in  equation  (i).     It  becomes 
a'—a(a  +  b)-^f=o,  (^) 

or  —ab-\-f=o,  (6) 

whence  f=(^b.  (-]) 

That  is.  the  absolute  term  is  equal  to  the  product  of  the  two 
roots. 

(b).  If  the  two  roots  equal  each  other,  that  is,  if  each  is  equal 
to  «,   the  form  (x—a)(x—b)=o  becomes  (x—a)(x—a)—o,  or 


Theory  of  Quadratics. 


69 


^'''—2ax-\-a^=o,  where  it  is  seen  that  —2a  is  the  sum  of  two  roots 
with  the  opposite  sign,  and  a^  equals  their  product. 

II.  Examples.  We  can  now  form  a  quadratic  equation  which 
shall  have  any  two  roots  we  desire.  Suppose  we  wish  a  quadratic 
whose  roots  shall  be  3  and  5.  Then  ^=—(^3-1-5;=— 8,  and 
/=3  X  5=  15-     Then  the  equation  is 

x^~8x-\- 15=0. 
/.     Form  the  equation  whose  roots  shall  be        3  and  —5. 
2.         ''       "  "  "         "  "       -3  and      5. 

J.         "       "  ''  ''         "  "       —3  and —5. 

a 

5.  "       "  **  "         "  "  2-f  ^3and2  — n/3 

6.  "       "  "  "         "  "     Vyand— Vy- 

7.  *'  "  "  "  "  "  —5  and  o. 
S.  ''  ''  "  "  "  "  6  and  6. 
p.         "       "           "             "         "             "          o  and  o. 


12.  The  student  should  not  understand  that  there  is  only  o?ie 
method  of  solving  the  quadratic  equation.  The  fact  is  that  the 
result  may  be  reached  in  a  great  variety  of  ways,  that  of  IV,  Art. 
9,  merely  being  one  among  a  great  number.  But  many  of  the 
different  methods  that  have  been  proposed  are,  in  the  last  analysis, 
essentially  the  same,  and  they  all  resolve  themselves  into  the  one 
principle  of  reducing  the  quadratic  to  some  form  of  a  simple  equa- 
tion. We  give  a  few  methods  of  solution  to  show  the  student 
what  a  variety  of  means  may  be  made  use  of  in  such  work. 

(a).     By  reduction  to  an  incomplete  quadratic. 

x^-\-ex-i-/=o. 
Suppose  x=:j/—^e,  where  j  is  a  new  unknown  quantity  ;  then 
the  equation  becomes 

or  y--e}'+ie'-hej-ie'-i-/=o, 

or  y—}^r-{-/—o. 


70  Algebra. 

This  is  an  equation  of  the  first  degree  in  terms  of  j'\  Solving 
we  obtain 

whence  j=  ±  V  \e'  — /, 

and  since  x=^r—\e, 

Solve  in  this  manner  the  equation  x^^^x—i^=o. 

(b).  By  consideriyig  the  quadratic  as  the  product  of  two  Imear 
factors. 

Suppose  the  function  of  x  to  be  the  product  of  two  factors  of 
the  form  (x-\-\e-\-2i)(x-\-\e—u),  where  u  is  a  new  unknown 
quantity.     Then  we  have  the  equation 

x'-\-€X-\-f=^(x-\-\e-\-u)(x-\-\e—u). 
Expanding  the  right  member  of  the  equation  we  obtain 

x:'  +  ex +/=  x'-\-ex-\-  \e^ —u^. 
Therefore  u'=\e'—f, 

whence  7i=d^s/  ^e'  —f 

Now  as  the  product  of  the  two  factors  (x-{-^e-^u)(x-^\e—zi) 
must  equal  zero  we  must  take  x  either  —^e^-u  or  —\e-]-ti.  Take 

the  former,  and  

.r=  —\e—  ?/=  — |^=F  V  i^'— /. 

Solve  b}^  this  method  the  equation  .^:''—6-^■=I6. 

(c).     By  the  sum  and  product  of  the  roots. 

Suppose  the  two  roots  of  x''-\-ex-^f=^o  to  be_>'  and  z.    Then  we 

know  by  Art.  lo, 

y-\-z=^—e  (i) 

and  y^=f^  (^) 

squaring  (i)  we  obtain 

_>'"  +  2y2  -\-z^=e\  r  3  j 

Subtracting  four  times  (2)  from  this 

y^-—2yz-^z'=e'—4f 

or,  extracting  the  root,    r— ^=±V^ — 4/i     * 
and  since  y-^z=—e, 

—e±\/e^—4.f 
J'=~ — T ■ 


2 
Solve  in  this  manner  the  equation  3:1" — 5.^+2=0. 


Theory  of  Quadratics.  71 

13.  Discrimination  of  the  Roots  of  the  Quadratic  Equa- 
tion.    The  roots  of  the  equation 

:x^-hex-\-f=o 

are  x=—^e-h^/{e'—/and  x^—\e—sj\e''—f. 

(a).     \i\e'—f  is  positive  there  are  two  real  and  unequal  roots. 

(b),     \i\e—f\s  negative  there  are  two  imaginary  roots. 

(c).  If  \e'—f  is  zero  the  two  values  of  x  each  reduce  to  —\e 
and  the  two  values  oi  x  are  real  and  equal. 

(d).  \i\r—f  is  a  perfect  square  the  two  roots  are  rational,  if 
e  is  rational. 

(e).     If  \e-—f\s>  not  a  perfect  square  the  roots  are  irrational. 

The  expression  \e''—f  is  called  the  Discriminant. 

The  case  where  \e^—f  is  zero  deserves  further  attention.  If 
J(?^— y=o  then  \e'=^f  and  the  equation  ji--f^-^'+y==o  becomes 

x^-\-ex-i-\e'=o 
or  (x-\-^e)(x-hie)=o. 

Whence  we  see  that  when  a  quadratic  equation  has  two  equal  roots 
the  function  of  x  is  a  complete  square. 

14.  To  Find  the  Conditions  that  a  Quadratic  Equation 
MAY  HAVE  TWO  POSITIVE  RooTS.    Represent  the  roots  by  a  and  d. 

Then  since  —(a-\-b)^=e 

if  the  roots  are  both  positive  the  coefficient  of  x  must  be  negative. 
Also  since  CLb=f 

if  the  roots  are  both  positive  the  absolute  term  must  be  positive. 

Hence  the  full  co7idition  that  both  the  roots  of  a  quadratic  be  posi- 
tive is  that  the  coefficient  of  x  be  negative  and  the  absolute  term 
positive. 

15.  To  Find  the  Condition  that  a  Quadratic  Equation 
MAY  HAVE  Two  Negative  Roots.     Represent  the  roots  as  before. 

Then  since  — (a-{-b)^=e 

if  both  roots  are  negative  the  coefficient  of  x  must  be  positive. 
And  since  (ib=f 

if  both  roots  are  negative,  the  absolute  term  must  be  positive. 

Hence  the  full  condition  that  both  the  rootsofaqimdroHc  be  neg- 
ative is  that  the  coefficieiit  of  x^be  positive^id  the  absolute  ternj/  we^ 


72  Algebra. 

16.  'To  Find  the  Condition  that  a  Quadratic  Equation 
MAY  have:  onk  Positive  and  one  Negative  Root. 

Since  ^^=/ 

if  the  roots  are  of  opposite  signs  the  absolute  term  must  be  negative. 

Since  —(a-{-d)  =  e 

if  the  positive  root  is  numerically  the  greater,  e  is  negative  and  in 
case  the  negative  root  is  numerically  the  greater,  e  will  be  pcsitive. 

The  conditio7i  that  a  quadratic  have  roots  of  opposite  signs  is  merely 
that  the  absolue  term  be  negative,  but  if  the  coefficient  of  x  is  nega- 
tive the  positive  root  is  numerically  the  greater  and  if  the  coeffi- 
cient of  X  is  positive  the  negative  root  is  numerically  the  greater. 

17.  BxAMPivES.  Discriminate  the  roots  of  the  following  equa- 
tions; that  is,  tell  by  inspection  whether  the  roots  are  real  or  im- 
aginary, and  if  real,  tell  whether  they  are  positive  or  negative. 

I.     x^'-f-Sjc— 9=o. 


-T^ -f  7 O-T -f- 1 2 GO = O. 

:r^— 4Jt-+4=o. 
x^-{-iox-\-/^^=o. 

X'^  —  ^X-\-20  =  0. 

x''=iojt"— 25. 

X^—\2X=  —  2^. 


18.  In  a  manner  similar  to  that  of  Arts.  14 — 16  the  student 
may  determine  the  following  : 

1.  Find  the  condition  that  a  quadratic  equation  may  have 
two  roots  numerically  equal  but  of  opposite  signs. 

2.  Find  the  condition  that  a  quadratic  equation  may  have 
two  roots  which  are  reciprocals  of  each  other. 

J.     Find  the  condition  that  a  quadratic  equation  may  have 
one  root  equal  to  zero. 

19.  M1SCE1.1.ANEOUS  Exercises  in  the   Theory  of  Quad- 
ratics. 

I.     If  a  and  d  are  the  roots  of  x^-\-ex-^f=o,  find  the  value 
of  ^^-f-<^^  in  terms  of  <?  and/. 


Theory  of  Quadratics. 


73 


whence 

and 

Therefore 


a-\-d=—e 

ad=f 

a:'-\-2ab-\-b^=r 

2ab=2f. 

I    .  I 


Find  the  vahie  of  '  -\-^^  in  terms  of  e  and  f. 
a-\-b         c 


a      b 


ab 


3- 
4- 


Prove  (a  —  b)'=e'~^f. 


Find  the  value  of  r+  -in  terms  of  e  and  /. 
b     a  -' 


5.     Given  the  equation  x" + ex  +/=  o,  form  the  equation  whase 
roots  are  the  squares  of  the  roots  of  this  equation. 

If  the  roots  of  this  equation  be  called  a  and  b,  the  roots  of  the 


and  b". 
a'-\-b' 
e^-2f. 


The  coefficient  of  ;*:  must 


required  equation  will  be  d' 

then  be 

or,  by  Ex.  i, 

The  absolute  term  must  be 

or 

Hence  the  required  equation  must  be 

6.  Form  an  equation  whose  roots  shall  be  the  reciprocals  of 
the  roots  of  x^-^ex-\-f^=^o. 

7.  Prove  that  the  equation  x^—kx—d^=o  cannot  have  im- 
aginary roots. 

8.  Find  the  value  of  7n  such  that  the  roots  oi  x^-\-ex-\-m—o 
will  differ  by  2. 

A— 9 


CHAPTER  VI. 

SINGIvE    KQUATIONS. 

I,  Every  equation  containing  one  unknown  quantity  can  be 
put  in  the  form 

Functio7i  of  x= o 

by  transposing  all  the  terms  to  the  left  side  of  the  equation. 

If  it  is  an  equation  of  the  first  degree  it  will  always  reduce  to 
the  form 

.1"  —  «  =  o, 

where  a  must  stand  for  any  quantity  whatever,  positive  or  nega- 
tive, integral  or  fractional,  commensurable  or  incommensurable. 
It  is  evident  that  this  equation  has  the  root  a  and  no  other.  An 
equation  of  the  first  degree  might  be  defined  as  an  equation  which 
can  be  placed  in  the  form  of  a  rational  integral  linear  function  of 
X  equal  to  zero. 

We  have  seen  that  every  quadratic  equation  can  be  placed  in 
the  form 

(x—a)(x—b)=o, 

which  has  the  two  roots  a  and  b  and  no  others.  Thus  every 
quadratic  equation  can  be  placed  in  the  form  of  the  product  of 
two  rational  integral  linear  functions  of  x  equal  to  zero. 

It  will  be  proved  in  Part  II  that  every  cubic  equation  can  be 
put  in  the  form 

(x—a)(x—b)(x—c)=-o, 

and  that  it  has  three  roots,  a^  b,  and  c,  and  no  others.  That  is, 
every  cubic  equation  can  be  placed  in  the  form  of  the  product  ot 
three  rational  integral  linear  functions  of  x  equal  to  zero. 

It  will  also  be  shown  that  an  equation  of  the  fourth  degree  can 
be  thrown  in  the  form 

(x — a)(x — b)(x — c)(x — d)=-o. 

These  and  other  important  properties  of  equations  containing 
qUc  unknown  quantity  were  first  discovered  by  Vieta  (1540  — 
1603),  but  were  independently  and  more  elaborately  treated  by 
Harriot  (1560 — 162 1). 


SiNGi^K  Equations.  75 

2.  We  are  led  to  inquire  what  operations  can  be  perfonned 
upon  the  members  of  an  equation  without  modifying  the  values  of 
the  unknown.  Now,  by  the  principles  of  algebra,  mi  equation 
remains  true  if  we  unite  the  same  quantity  to  both  sides  by 
addition  or  subtraction  ;  or  if  we  multiply  or  divide  both  mem- 
bers by  the  same  quantity  ;  or  if  like  powers  or  roots  of  both 
members  be  taken.  But,  as  hinted  in  IV,  Art.  2,  these  operations 
may  affect  the  value  of  the  tinknoivn.  Thus  the  roots  of  the 
equation 

3r-^— 5y>=-^Y-^"— 5^*  +  -^"— 25  (I) 

are  —  i  and  5.  Either  of  these  when  substituted  for  x  will  satisfy 
the  equation.  But  divide  the  equation  through  by  x— 5.  The 
resulting  equation  is 

3=.v  +  .r-f5.  (2) 

Now  this  equation  is  not  satisfied  for  x=z^.  The  sole  root  is 
—  I.  Hence,  although  equation  (2)  mUvSt  be  t^-ue  if  (i)  is,  yet  the 
equations  are  not  equivalent,  since  their  solutions  are  not  iden- 
tical. One  root  has  disappeared  in  the  transfonnation.  Just  how 
this  occurs  will  be  best  seen  after  we  place  (i)  in  the  form 
(x—a)(x—b)  =  o.  Since  the  roots  of  (i)  are  —  i  and  5,  by  the 
principle  of  V,  Art.  6  it  is  equivalent  to 

(x-s)(x+i)=^o.  (3) 

Now,  if  we  divide  this  through  by  -^"—5,  we  remove  that  factor 
in  the  left  member  which  is  zero  for  -r=5.  Consequently  the 
equation  will  be  no  longer  satisfied  for  .1  =  5.  If  we  should  divide 
through  by  x-\-i  the  equation  will  be  no  longer  satisfied  for 
x=  —  I . 

Also  consider  the  equation 

.1-"— 6.r+8=o.  (4) 

It  is  satisfied  for  .1  =  2  or  -f=4.     Now  multiplying  both  members 

t>y  -^"  +  3  we  obtain 

(x+:,)(x--6x-\-^)=o.  (5) 

But  this  equation  is  satisfied  for  either  .v=— 3,  or  .1  =  2,  or  .1  =  4. 
Hence,  although  multiplying  both  members  of  (4)  by  .i"-{-3  has 
not  altered  the  equality,  yet  a  value  of  x  extraneous  to  the  orig- 
inal equation  has  been  introduced. 

Again  the  equation 

2-r— i=.i--j-5  (6) 


76  Al^GKBRA. 

is  satisfied  only  by  the  value  -r=6.  Now  square  both  sides  of  the 
equation,  obtaining 

4Jt-^— 4.r-j- 1  =A-=-|-  io.r+  25,  (7) 

which  is  satisfied  for  either  x=6  or  x=—\.  Here,  obviously,  an 
extraneous  solution  has  been  introduced  by  the  operation  of 
squaring  both  members. 

In  a  like  manner  notice  the  effect  of  taking  a  root  of  both 
members  of  an  equation.     Thus  suppose 

^■^=(x—6)\  (8) 

This  is  satisfied  for  either  x=2  or  —6.  Take  the  square  root  of 
each  member  and  we  obtain 

2x=--x—6,  (9) 

which  is  satisfied  only  by  x=—6.  We  have  lost  one  of  the  solu- 
tions of  the  equation  during  this  transformation,  Equation  (?>) 
is  really  not  equivalent  to  (()),  but  to  the  two  equations 

^2X=  +  (x-6)\  . 

{2X=-(x-6)   \  ^^^^ 

We  have  given  examples  enough  to  show  that  certain  opera- 
tions upon  an  equation  may  modify  the  solution.  Thus  we  see 
that  during  a  series  of  transformations  which  sometimes  an  equa- 
tion must  undergo  before  we  can  reach  the  values  of  the  unknown 
it  is  possible  that  the  solutions  that  satisfy  the  original  equation 
may  all  be  lost  and  that  any  number  of  new  ones  may  be  intro- 
duced, so  that  the  final  results  may  have  no  relation  at  all  to  the 
problem  in  hand.  It  is  now  proposed  to  formulate  certain  propo- 
sitions which  will  enable  us  to  tell  the  exact  place  in  the  process 
of  any  solution  where  roots  may  be  lost  or  new  ones  may  enter. 
We  will  then  be  able  to  perfonn  the  different  operations  on  the 
members  of  an  equation  if  we  will  note  at  the  time  their  effect  on 
the  solution  and  finally  make  allowance  for  it  in  the  result.  This 
fact  must  be  emphasized  :  ^/le  test  fo?-  a?ty  solution  of  aii  equation 
is  that  it  satisfy  the  original  equation.  ''  No  matter  how  elaborate 
or  ingenious  the  process  by  which  the  solution  has  been  obtained, 
if  it  do  not  stand  this  test  it  is  no  solution  ;  and,  on  the  other 
hand,  no  matter  how  simply  obtained,  provided  it  do  stand  this 
test,  it  is  a  solution." — Chrystal. 

When  one  equation  is  derived  from  another  by  an  operation 
which  has  no  effect  one  way  or  another  on  the  solution,  it  may  be 


SiNGivE  Equations.  77 

spoken  of  as  a  legitimate  transformation  or  derivation  ;  when  the 
operation  does  have  an  effect  upon  the  final  result,  it  may  be 
called  a  questionable  derivation,  meaning  thereby  that  the 
operation  requires  examination. 

If  there  are  two  equations  such  that  any  solution  of  the  first  is 
a  solution  of  the  second,  and  also  that  any  solution  of  the  second 
is  a  sokition  of  the  first,  the  two  equations  are  said  to  l)e 
equivalent. 

3.  Theorem.  The  trayisformatioii  of  an  equation  by  the  addition 
or  subtraction  from  both  members  of  either  a  kyiown  quantity  or  a 

functio7i  of  the  unknown  is  a  legitimate  derivation. 

An  equation  containing  one  unknown  quantity,  as  it  commonly 
appears  with  quantities  on  each  side  of  the  equation,  may  be 
generalized  in  thought  by  the  expression 

A    function  of  x=  Another functioti  of  x. 
Or,  using  L  to  represent  the  left-hand  side  of  the  equation,  what- 
ever it  may  be,  and  R  to  represent  the  expression  on  the  right- 
hand  side,  we  can  represent  any  equation  very  conveniently  by 

L  =  R.  (I) 

Now  suppose  that  T,  which  ma}^  be  either  a  known  quantity  or  a 
function  or  the  unknown,  be  added  to  both  members  of  the  equa- 
tion, making 

L+7  =  R-\-T.  (2) 

Now  it  is  plain  that  (2)  cannot  be  satisfied  unless  L=R  and  that 
it  is  satisfied  if  L  —  R.  Hence  (2)  means  no  more  nor  less  than 
(i).     Therefore  the  derivation  is  legitimate. 

4.  CoROLivARY.  Transposition  of  terms  from  one  member  to 
the  other,  changing  the  si^ns  at  the  same  time,  is  legitimate.  Thus 
\i  L  —  R,  to  pass  to  L—R=o  is  merely  subtracting  A'  from  both 
members. 

5.  Theorem.  Multiplying  both  members  of  an  equation  by  the 
same  exp^rssion  is  legitimate  if  the  expression  is  a  known  quantity, 
but  questionable  if  the  expression  is  a  function  of  the  unknown. 


78  Algebra. 

Represent  the  equation  by 

L  =  R.  (I) 

Multiply  both  members  by  T,  obtaining 

LT=RT.  (2) 

Now  this  may  be  written  as 

(L-R)T^o.  (3) 

If  T  is  a  known  quantity  this  can  only  be  satisfied  by  the  sup- 
position that  L  =  R,  that  is,  the  equation  is  equivalent  to  (ij. 
But  if  7^ is  a  function  of  the  unknown  (for  example,  2x  or  A-I-5, 
or  -r^+8)  then  (3)  may  be  satisfied  by  any  value  of  the  unknown 
that  Will  make  7^=o  (such  as  x=o,  or  ^  =  5,  or  ,f=  — 2,  respect- 
ively, in  the  three  examples  given),  whence  (3)  would  not  be 
equivalent  to  (i)  but  to  the  two  equations. 

\  L  =  R} 
\  T=o.  ) 

6.  Corollary.  //*  any  equation  involves  fraetions  7vith  only 
know)i  qnantities  in   the  denominators,   it  is  legititnate  to  elear  of 

fractions.     The  multiplier  in  this  case  is  a  known  quantity. 

7.  ThEORKM.  It  a^i  eqiiation  involves  irreducible  fractions  zvith 
unknown  quantities  in  the  denominators,  and  the  denominators  are 
all  prime  to  each  other,  it  is  legitimate  to  integralize  by  multiplying 
through  by  the  least  common  multiple  of  the  denominators. 

To  illustrate  the  reasoning  take  the  equation 

I      2      3 

where  the  fractions  are  supposed  to  be  in  their  lowest  tenns  and 
X ^,  X ^,  X^  represent  diffei^ent  functions  of  the  unknown  and 
where  A,  B  and  C  are  either  known  quantities  or  functions  of 
the  unknown.  Multiplying  by  the  least  common  multiple  of  the 
denominators  we  obtain 

AXX^-^BXX^-^CX^X^^o,  (2) 

Now,  since  X^ ,  X^  and  X^  are  prime  to  each  other  no  common 
factor  has  been  introduced  by  multiplying  by  X  X^X^,  and  con- 
sequently no  additional  solutions  can  appear. 


Single  Equations.  79 

8.  As  an  example  under  the  above  theorem  take  the  equation 

II  — 2Jl-       3-V— I 

These  fractions  are  in  their  lowest  terms  and  their  denominators 
are  prime  to  each  other.  The  least  common  multiple  of  the  de- 
nominators is  f  II  — 2.rX3-v— I  j.  Multiplying  through  by  this 
we  obtain 

(^X—l)('J  —  x)  +  (ll  —  2X)(4X~s)=2(ll—2X)(T,X—l).     (2) 

Now  we  can  see  that  although  ( i )  has  been  multiplied  through 
both  by  (i\  —  2x)  and  (2>^—i),  yet  neither  of  these  has  been  in- 
troduced as  a  factor  through  the  equation.  Hence  there  is  no  ad- 
ditional solution  introduced.  The  roots  of  (^2J  will  in  fact  be 
found  to  be  4  or  —10,  which  values  also  satisfy  (\). 

But  an  extraneous  solution  may  be  introduced  if  the  denomin- 
ators are  not  prime  to  each  other,  or  if  some  of  the  fractions  are 
not  in  their  lowest  terms.     Thus 

has  two  denominators  alike,  and  consequently  not  prime  to  each 
other.  Multiplying  through  by  the  common  denominator  x^—g, 
we  obtain 

^x(x  +  2.)  =  6(x-^)^^(x+2>)  (4) 

or,  reducing,  .1'— 2.1=4  (s) 

whose  roots  are  3  and  —  i.  Now  if  we  put  the  original  equation 
(t,)  in  the  form 

X— 3      x-\-3 
that  is  3=     ,  (6) 

it  is  seen  that  it  is  satisfied  only  for  .1  =  — i.  Hence  a  solution 
was  introduced  in  clearing  (t,)  of  fractions.  It  is  easy  to  see  that 
(t,)  is  really  equivalent  to  (6)  and  hence  that  in  clearing  (t,)  of. 
fractions  by  multiplying  by  .r-— 9  we  multiplied  by  .v— 3  when  it 
was  not  necessar\^  ;  this  is  where  the  solution  .r=3  was  introduced. 

9.  Theorem.       /szrrr  equation  can  be  inte^ralized  kiritimately. 
For  if  the  several  fractions  in  the  equation  are  not  in  their  low- 
est terms  thev  can   be  so  reduced.     Then  these  fractions  can   all 


8o  AlyGKBRA. 

be  transposed  to  one  side  of  the  equation,  their  common  de- 
nominator found  and  then  added  together.  This  will  now  give 
but  one  fraction  in  the  equation,  and,  when  this  is  reduced  to  its 
lowest  terms,  we  will  have  an  equation  of  the  form 

N 
which,  since  -  is  in  its  loivest  terms  by  supposition,   will  take   on 

no  additional  solutions  when  multiplied  through  by  D^  according 
to  Art.  7. 

10.  Theorem.  The  raishig  of  both  members  of  an  equation  to 
the  same  poiver  is  equivalent  to  7nultiplying  throni^h  by  a  funetiofi 
of  the  unk7ioum  ayid  hence  is  a  questionable  derivation. 

Take  the  equation 

7.  =  ^  (1) 

and  raise  both  members  to  the  «th  power,  obtaining 

IJ'^R".  (2 

Now  r  O  is  equivalent  to 

L-R^o 
and  (2)  is  equivalent  to 

But  (^)  can  be  derived  from  (^3  j  by  multiplying  both  members  by 

/."-'+Z«-=7?-f-/."-^y?'--f .  .  .-^L'R"-'^LR'-'+R"-' 

wdience  f  2  j  is  equivalent  to  the  t7co  equations 

(  L  =  R  ) 

{L"-'  +  L"-'R-\-L"-'R'-  +  .  .  .^L'R"-^  +  LR"-'  +  R"-'  =  o.  \ 

11.  Theorem.  Dividing  both  members  of  an  equation  by  the 
same  expression  is  legiti7nate  if  the  expression  is  a  knoum  quantity, 
but  questionable  if  it  is  afunctio7i  of  the  imknoum.\ 

Suppose  both  members  of  the  equation  to  be  divisible  by  T  and 
write  the  equation 

LT^RT.  (i) 

Now  if  T'ls  a  known  quantity,  then  by  Art.  5  this  equation  is 
equivalent  to 

L=R  (2) 

whence  division  by  7^  w^ould  be  legitimate.  But  if  7"  is  a  func- 
tion of  the  unknown  quantity,  then  (i)  \s  equivalent  to  the  two 

equations  ^  L=R 

T=o. 


SINGI.E  Equations.  8i 

Division  by  T  would  give  us  but  one  of  these,  and  consequently 
solutions  would  be  lost.  Hence  the  division  by  a  function  of  the 
unknown  is  a  questionable  derivation. 

12.  Theorem.  The  extraction  of  the  same  root  of  both  members 
of  an  equatio7i  is  equivalent  to  dividing  by  afnnction  of  the  unknown 
and  hence  is  a  questionable  derivation. 

For  we  can  pass  from 

to  L=^R  (2) 

by  dividing  both  members  of  ("i^  by 

L"-'-^L'-^R-JrL"-'R'-\-  .    .    .  ^L^R"-^-JrLR"-^^R"-\ 
Hence,  by  Art.  11,  root  extraction  is  a  questionable  derivation. 

13.  Examples  of  the  Integrauzing  of  Equations.  In  the 
following  equations  the  student  should  note  the  precise  effect  of  all 
questionable  operations  at  the  time  they  are  performed. 

I.     Solve  \(x-^)(x-^)(x-2)==(x~^)(x-^2>)(^+2). 
X -h  2      X—  2_5 
X—  2       X  4-  2       6' 


2.     Solve 


J.     Solve   --i    -  +  ^l-  =  i. 

12     x-^i 

4.  Solve  -H — = . 

X       X  X 

c^    -  X'^  XI 

5.  Solve-— ,=  i 

6.  SolvL  i =   ^. 

x—i         X         6 


7.     Solve 


fjf2£--  i_5  ^  -r  +  5 
x'—x — 6        x-\-  2' 


14.  Examples  of  the  Rationalization  of  Equations.  The 
most  expeditious  method  for  rationalizing  any  given  equation  de- 
pends upon  the  peculiar  make  up  of  the  equation,  and  can  only 
be  determined  by  the  student  after  a  little  experience  with  this 
class  of  equations. 
A— 10 


82  Algebra. 

/.     Giveji  \/g^-  x+x==ii.  (i) 

Transpose  ever>'thing  but  the  radical  to  tbe  right-hand  side  of  the 
equation  and  we  obtain 

\/g  +  x=\\~x.  (2) 

Squaring  both  sides  gives 

9  +  -i'=i2i  — 22.A-I-X'  (t^) 

and  solving  this  quadratic  we  find 

x=7  or  16. 
From  (2)  to  (t^)  is  a  questionable  derivation  ;  for  squaring  both 
members  of  an  equation,  L=^R,  we  have  found  ("Art.  10)  to  be 
equivalent  to  multiplying  through  by  L-\-R,  and  that  the  result- 
ing equation  is  equivalent  to  the  two  equations 

Therefore  (^3  j  is  equivalent  to  the  two  equation 

f 
\ 

i 

or  to 


s/  (^-\-x—\\—x  1 
\/9  +  -^H-ii--^=o  I  ^^^ 


j  — V9+-^^+-^=ii)  ^^^ 

Hence,  if  we  understand  equation  (\)  X.o  read 

The  positive  square  root  of  (<:)-\-x)-\-x=^\\ 
then  a  new  solution  has  been  introduced  between  (2)  and  ('3J. 
But  if  we  understand  equation  (\)  \.o  read 

A  square  root  of  (g-{-x)-\-x=  1 1 
then  it  is  equivalent  to  both  the  equations  in  (5),  and  no  solution 
has  been  introduced.     This  is  because  the  introduced  equation, 
±/.-f /v'=o  is  identical  with  the  original  equation  :^L—R. 

In  these  cases  the  student  will  always  find  that  ratioyialization 
may  or  may  not  be  considered  as  a  questionable  derivation  according 
us  we  consider  the  radicals  to  call  for  a  particuIvAR  root  or  AiSiY  root 
of  the  expressions  involved. 

It  is  more  in  accordance  with  the  generalizing  spirit  of  algebra 
to  consider  the  radical  sign,  wherever  it  occurs,  as  calling  for  any 
of  the  possible  roots.  This  will  be  better  appreciated  by  the  stu- 
dent when  he  learns  in  Part  II  that  every  expression  has  three 
different  cube  roots,  four  fourth  roots,  five  fifth  roots,  etc. 


SiN(;i^E  KquatioNvS.  S3 

2.     Solve  \/-*-+V-r  +  6=3.  (i) 

Squaring  each  side  of  the  equation,  obtain 

.r-f  2V^'-i"6i+xH-6=9.  .(2) 

Transposing    all    but    the    radical    to    the  right-hand    side    this 
becomes 

Squaring,  we  obtain 

4-1"+  24.r=9—  1 2Jf +4.r  (/^) 

or  -^=i- 

What  are  the  questionable  steps  ?     What  is  their  effect  ? 
The  above  solution  is  really  equivalent  to  the  following  : 
\/ x-\-  sj X  -f-  6=3. 
Transpose  the  3  to  left  member,  obtaining 

V -^ ■+ V -^  +  6— 3=0 ; 
Multiplying  through  by  the  rationalizing  factor  of  left  member, 
(III,  Art.  26)  we  obtain 

Wx-V  \/^rT6-3)(  V-^+  V.r  +  6+3) 

x(V-i^— V.v"-f  6— 3)(\/^— V-^t;-{-6-f  3)=o, 
which  reduces  to 

.4.1-— 1=0.  (5) 

or  ^  =  4- 

The  introduced  equations  arc 


V.r-f- \/x  4-6  +  3—0        ; 

\/-^"— V-^' -I- 6— 3=0 

V -^'— V -^ -f  6  +  3 = o. 
Here,  then,  is  an  apparent  paradox  :    three  solutions  seem  to 
have  been  introduced,  yet  there  is  only  ^;?^  in  all  ! 

This  can  be  explained  in  the  following  manner.  If  we  regard 
the  radical  signs  as  calling  for  any  one  of  the  two  roots  of  the  ex- 
pression underneath,  then  the  introduced  equations  are  all  iden- 
tical with  the  original  equation  ana  hence  could  not  give  rise  to 
a  differe7it  solution.  If  we  restrict  ourselves  to  using  that  square 
root  in  each  case  which  has  the  sign  given  before  the  radical, 
then  none  of  the  introduced  equations  have  any  solution  Tvhatever^ 
and  hence  no  solution  is  introduced  in  this  case. 


^  .       s/ x-\-s/  a-\-x 

3.     Solve  —^ ^^  _^.=.=.v. 

sJ X—  sf  a  -\-  X 


^4  Algebra. 

Sometimes  such  equations  are  best  simplified  by  first  rationaliz- 
ing either  the  numerator  or  denominator  of  the  fraction.  Ration- 
alizing the  denominator  of  this  fraction,  the  equation  becomes 

2 -r -j- <2  +  2  V  ^-^' 4- ■^^_ 
X — a — X  ' 

or  2sJ  ax-\-x^^=  —  (ax-\r2x-\-a), 

whence  \ax  ■\-  ^x'  =(ax-\-2x-\-a)\ 

etc. 
Solve  .r-f-  s/  X  -\-  3=4-^' —  i . 

Solve     \/'l4  —  X-\-  ^/  II  —  Jf= ^-rr^=r- 

V 1 1  —  -^ 

Solve  s/ ^x -^  g— s/ X — i=  \f  x-\- 6. 

y.     Solve  V-^  — 9+ V-^ +  12=  V-^  — 4+ V-^  4- 13. 

<y.     Solve  sj  x^a^ — ^/  x=b. 

^  ,  20X  .      18 

Solve V  10.^—9= —  +9. 

Vio;r  — 9  Viojr  — 9 

vSolve  V6jr+4  +  v/^~*"+io^-f"3.^-f7^=-^'f3- 

Solve  -^  +  ^  V4^'  -T^^  -  4  V4^-V 

6  +  5V4-^  — 7     3— ioV4-^— 7 

^,  ,       Vi+x^+^i-xf' 
Solve     -  -^^ ^— rr. — =  I. 

Rationalize  ^.r''— \/-^'^=o. 
Solve -^^+^^-3_     3 


// 

J2 


sj  x—sj  x—-T^     ■^— 3- 
Criticise  the  following  solution  : 

Multiply  both  terms  of  the  first  fraction  by  sj x-\-  \/ x—-^,  and 
we  have 

x—(x—2,)         x—^ 

or  .  ..    (V^+V-^-^T=-^   .  r3/> 

.r     3 

Extracting  the  square  root, 

\/^+V^-3=    ,^       .  (\) 

V-^"— 3 


Single  Equations.  85 

Clearing  of  fractions, 

V^"— 3-^"+-^-3=3  ;  (5) 

whence  x^^  (6) 

2  2 

Tfy.     Solve  .1== 1 --  ^. 

X  -^  Sf  2-\-X'      X—\/2-hX' 

16.     Solve  \/ X -}-  \/  2 -\- X 


V2  +  X 

//.     Solve  V  5-^  +  10=  V  5-^+ 2. 
iS.     Solve  \/i+-^-+ ViH-;ir  +  >/i+.r=  Vi— -^■. 

/p.     Solve -^—-~=(x-\-2)\ 

X — V -^ — 9 
Rationalize  the  denominator  of  the  fraction. 

15.  Equations  which  can  be  Solved  as  Quadratics.  The 
method  employed  in  the  solution  of  quadratic  equations  will  some- 
times enable  us  to  solve  equations  of  other  degrees,  or  even  irra- 
ttonal  equations.     Thus  consider  the  equation 

3-^— 5-^'  +  4=2.  (i) 

Multiply  both  members  by  four  times  coefficient  of  .x"*  and  add  the 
square  of  the  coefficient  of  x'  to  each  side,  as  in  the  solution  of  a 
quadratic  equation.     It  then  becomes 

36^1:^— 6ojt:''-f25=r.  (2) 

The  left-hand  member  is  now  a  perfect  square.  Whence,  extract- 
ing the  square  root  of  both  members,  equation  (2)  becomes 

6jr"— 5==ti 
whence  x^==  i  or  |. 

Therefore  x=  -f  i  or  —  i  or  -f  Vl  or  —  V|  • 

As  another  example  consider  the  equation 

4\/x  +3A-=4.  (i) 

Put  1'=  \/x,  whence  it  is  seen  that  (1)  becomes 

4.^+3/'— 4-  (^) 

Solving  this  quadratic  we  find 

j'=|  or  — 2. 
Whence,  since  y=  \^  x 

\/-^  =  |  or  —2. 
Therefore  x==^  or  4. 

These  examples  suggest  the  following  theorems  : 


86  Algebra. 

16.  Theorkm.  ."/;/>'  equatio7i  ivhich  can  be  placed  in  the  form 
x'^"-\-ex"-\-f=o  can  be  solved  as  a  quadratic. 

x'^''-\-ex"-\-f=o  (\) 

may  be  written 

which,  if  we  regard  (x")  as  the  unknown  qantity,  is  seen  to  be  in 

the  quadratic  form.     Completing  the  square  of  (2)  it  becomes 

(x'^-'^e(xn  -i-  \e^Y-f  r  3) 

Whence  x"-\-\e=  =b  sj  \e—f 

or  .r"  =  —  .}r  ±  V  \e  —f. 

1 

Therefore  x=( —\edti,s/ \e'-f)"  (a) 

which  is  the  solution  of  the  equation  of  the  proposed  form. 

17.  Theorem.  Ajiy  equation  ivhicli  can  be  placed  in  the  form 
X~  -^eX"-\-/=o,  tvhere  X  starids  for  arty  linear  or  quadratic 
function  of  the  unhioivn,  can  be  solved  as  a  quadratic. 

For,  by  the  last  article,  it  will  be  found  that 

1 

Now,  if  .Vis  a  linear  function  of  x,  this  equation  is  of  the  form 

1 

which  can  be  easily  solved  for  x. 

If  A' is  a  quadratic  function  of  r  equation  {\)  must  be   of  the 

form 

1 

ax'^bx-\-c={^-U±is/\r-fY  (^) 

Now  this  is  a  quadratic  equation  in  terms  of  x,  since  all 
other  quantities  in  the  equation  are  known,  and  hence  the  equa- 
tion can  be  solved. 

In  treating  examples  which  come  under  these  two  theorems  it 
may  be  possible  that  we  will  not  find  all  the  values  that  will  sat- 
isfy the  given  equation.  This  happens  because  we  are  not  always 
able  to  find  n  different  n  th  roots  of  a  quantity,  while  that  num- 
ber really  do  exist.     Thus  from  the  equation 

we  will  find  by  considering  x'^  the  unknown  quantity  that 

jt-3=27  or  —8 
whence  -^=3  or  — 2. 


Single  Equations.  87 

But  really  27  and  —8  have  each  ///r^d' different  cube  roots  instead 
of  merely  the  ones  we  have  written  above.  The  full  considera- 
tion of  this  matter  involves  subjects  vSomewhat  more  advanced, 
and  more  than  the  mere  statement  above  given  will  not  be  at- 
tempted until  Part  TI  of  the  present  work  is  reached. 

18.  Examples.  The  following  five  are  examples  under  the 
theorem    of  Art.    16  : 

/ .  Solve  x''  -f  1 6.V  ■  =225. 
2.  Solve  x""— ^x' +8=0. 
J .     Solve  6x* —35=11  X-. 

4.       Solve  x'-f|  =  J:^Ji:^ 
^ .       >^.t-3  —  2  V  X  -f  X=0. 

The  following  five  are  examples  under  the  theorem  of  Art.  17. 

6.     Solve  -^H-5V37— -^  — 43- 
Process  :  Subtract  37  from  each  side  of  the  equation,  obtaining 

-V— 37-+-5V37— ■''^— 6 
M^hich  may  be  written 

—  r37--v;  +  5V37--^-6 
or  r37--^>'-5V37--^"=-6. 

Putting  )'  for  \/t,7—x  this  becomes 


r=_y/=_6. 

Solving, 

j'==3  or  2. 

That  is 

V37— •^  — 3  or  2 

whence 

37— A-=9  or  4 

and 

.r=2S  or  33. 

The  same  example  may  be  treated  by  the  method  of  Art.  1 1 . 
7.     Solve  v^—V-^— 9=21. 
S.     Solve  2\/^'— 5-V+2— -^^"-f-8A-=3.v— 78. 

^.       Solve  (2X'—T,X-{-l)^=22X'''—T,;^X-\-  I  I. 

/o.     Solve  4x'—^x-\-20^/2x^—5x-i-6=6x-\-66. 
The  following. are  examples  of  either  the  theorem  of  Art.  16  or 
of  Art.  1 7  : 


88  AlvGEBRA. 

TT.    Solve  axV  x + 7^7^^ = c. 
y/  X 

12.     Solve  x~^—2X~^=%. 

2  i 

I  J.     Solve  Ji:"— 5^-"H-4=o. 

/^.     Solve  3-r— 20=7  V-^- 

75.     Solve  iox'^"-\-xn-\-24.=o. 
16.     Solve  T  T  ox~''  +1  =  21  x"^. 

ij.     Solve  sj x-\-^x  ^  =  5. 

18.  Solve  3.r"— 4x-f\/3-^'— 4^— 6=18. 

19.  Prove  that  the  equation  x^—97Ji'*+ 1500=204  is  equiva- 
lent to  the  equation  (x^—\b)(x^~%\)  =  o. 

1         -?        A 
^o.     Solve  2x'^  —  '}^x'^-\-x'^=^o. 

Result :  -r=o,  or  i,  or  8. 
21.     Solve  fjL— «)"  +  >—         r='^. 

^^.     Solve  8-v2"— 8.I.-  2«=63, 

i         I-         ± 
^j.     Solve  -r^  -f  8.r ^  +  g^r ^ =0. 


CHAPTER  VII. 

SYSTEMS   OF   EQUATIONS. 

1.  Definition.  If  a  number  of  equations  containing  several 
unknown  quantities  are  supposed  to  be  so  related  that  they  are 
all  satisfied  simultaneously  by  the  same  set  of  values  of  the  un- 
known quantities,  the  equations  are  said  to  constitute  a  5y.y/d'/w, 
or  a  System  of  Simultaneous  Equations. 

Thus  the  equations 

2-r-f  y-\-   5^=19"^ 

7^x-\-2y+   45'=  19  - 

are  satisfied  simultaneously  by  the  set  of  values, 

jr=i,jj/=2,  ^=3, 
and  are  said  to  constitute  a  system.  This  set  of  values,  or  the 
process  of  finding  them,  may  be  called  the  Solution  of  the  system. 
The  reader  is  supposed  to  be  already  familiar  with  methods  of 
solution  of  a  system  of  simple  equations  containing  as  many  equa- 
tions as  different  unknown  quantities,  such  as  the  system  given 
above.  The  systems  we  propose  to  consider  in  this  chapter  are 
those  involving  quadratics  or  equations  of  higher  degrees. 

2 .  The  student  should  not  suppose  that  every  system  of  equa- 
tions which  may  be  proposed  is  capable  of  solution.  It  is  one 
requirement  that  the  number  of  unknown  quantities  be  just  equal 
to  the  number  of  equations  in  the  system.  But  even  this  is  not 
all.  Some  of  the  equations  in  the  system  may  contradict  some 
of  the  others,  in  which  case  a  solution  is  impossible.  For  example, 
take  the  system 

34-2J/=2X  (l)\ 

X-  r=i  (2)\ 

From  equation  (2) 

x=  I  -hj'. 

Substitute  this  value  of  .r  in  equation  (\)  and  we  obtain 

3+2V=2  +  2J/, 

or  1=0, 


90 


Algebra. 


r=-2  J 


and  by  no  other  method  of  elimination  can  we  get  anything  but 
an  absurdity  from  the  given  system.  Equations  of  this  kind  are 
said  to  be  incompatible  because  one  equation  affirms  what  another 
denies.  We  will  see  this  to  be  so  in  the  above  system  if,  by 
proper  transformations  in  equation  (\),  the  system  be  written 

x—v=  I  (4)  \ 

These  equations  are  necessarily  contradictory  and  can  have  no 
solution. 

Another  example  of  an  incompatible  system  is 

x—y=4.     - 

X-^y  —  2:  =2      ) 

From  the  second  of  these  equations  it  is  seen  that 

Substituting  this  value  of  x  in  the  first  and  third  of  the  equa- 
tions in  order  to  eliminate  x,  we  obtain  the  system 

2J/— -2-=  — 

2_y—2 
and,  since  these  are  incompatible,  we  can  go  no  further. 

There  is  still  another  case  in  which  a  system  may  have  no 
solution.     Consider  the  equations 

4X=2>(2  —  3y)  j 
From  the  first  equation  we  find 

x  —  ^—^y. 
Substituting  this  value  of  x  in  the  second  equation  we  obtain 

which  reduces  to  0=0, 

and  we  get  no  solution.     Equations  of  this  kind  are  said  to  be 

dependent  because  the  equations  really  make  the  same  statement 

about   the    unknown   quantities.     This  will  be  seen  when,   by 

proper   transformations   in    the  equations,    the  above   system  is 

written. 

4-^4-97=6) 

4j»;-f  9j^/=6  j 
It  is  now  seen  that  the  equtions  of  the  system  do  not  state 
independent  truths,   and  consequently  the  system  has  no  more 
meaning    than    a    single    equation     containing    two    unknown 
quantities. 


Systems  of  Equations. 


91 


It  will  also  be  found  that  the  system 
4-^+ 3)'+ 22-=  I 

X-h7,(2-\-2)=2(l-y)) 

is  a  dependent  one,  the  dependence  being  between  the  first  and 
third  equations. 

We  may  then  enumerate  three  conditions  which  must  be  ful- 
filled by  a  system  of  equations  in  order  that  a  solution  may  exist : 

There  must  be  just  as  many  equations  as  there  are  unknown 
quantities. 

The  equations  must  be  compatible. 

The  equations  must  be  independent. 

3.  Of  course  if  any  equation  of  a  system  be  operated  upon  in 
any  manner  during  the  solution,  care  rnust  be  taken  that  the 
transformation  be  with  a  due  regard  to  the  theorems  in  VI,  Arts, 
3 — 12.  Obviously,  no  operation  which  it  is  questionable  to  per- 
form on  an  equation  standing  alone  can  be  legitimately  performed 
upon  one  belonging  to  a  system.  But  in  addition  to  the  reduc- 
tions which  single  equations  may  undergo,  equations  of  a  system 
permit  of  certain  transformations  peculiar  to  themselves,  and  it 
remains  to  investigate  the  possible  effect  of  these  on  the  solution 
of  the  system.  The  following  theorems  are  designed  to  point  out 
the  effect  on  the  result  of  the  ordinary  steps  in  the  process  of 
elimination. 


4-.  Theorem.     //  from  the  system  of  equatiotis 


L 


R, 


2i'e  derive  the  system, 


L=R, 
L.S  +  L,T=R.S  +  R,T 


(a) 


(b) 


U=R„ 

where  all  but  the  second  equation  rejnain  unchanged,  the  derivation 
is  legitimate  if  T  is  a  known  quayitity,  7iot  zero,  but  questionable  if 
T  is  a  function  of  the  unknown  quantities,  it  bei?ig  indifferent 
whether  S  is  a  hiown  qua7itity  or  a  function  of  the  unknown  OJies. 


92                                         Algebra. 

Write  system  (a)  so  that  it  will  read 

L -R  =o 

(I) 

L,-R,=o 

(2) 

L,-R„=o 

and  system  (^(^  j  so  that  it  will  appear 

L,-R  =o 

(3) 

S(L.-R,)-T(L.-RJ=o 

(4) 

(c) 


l„-r/ 

First,  suppose  T  a  known  quantity. 

Then  any  set  of  values  that  will  satisfy  (c)  must  make  L^— R,, 
L^ — R^,  .  •  •  and  Iy„— R„  each  zero.  But  any  set  that  makes 
these  zero  must  satisfy  (d)  also.  Hence  any  solution  of  {c)  is  a 
solution  of  (d). 

It  is  seen  from  ('3/ that  any  set  of  values  that  satisfies  (d)  must 
makely_  — R^  zero.     Equation  (^)  will  then  become 

TrL-RJ=o.  (s) 

Now  since  T  is  a  known  quantity,  not  zero,  this  cannot  be  sat- 
isfied unless  ly^— R^  is  zero.  Hence  any  set  of  values,  in  order  to 
satisfy  (d),  must  make  L^— R^  and  L^— R,  and  also  .  .  .  L„ — R« 
each  zero.  But  any  set  of  values  that  makes  these  zero  will  satisfy 
(c).     Therefore  any  solution  of  (d)  is  a  solution  of  (c). 

Now  we  have  shown,  Jirst,  that  any  set  of  values  that  will  satisfy 
(c)  will  satisfy  (d),  and  second,  that  any  set  of  values  that  will 
satisfy  (d)  will  satisfy  (c).   Hence  the  two  systems  are  equivalent. 

Second,  suppose  T  a  function  of  some  of  the  unknown  quanti- 
ties. 

In  this  case  equation  (^)  may  be  satisfied  by  any  set  of  values 
that  will  satisfy  the  equation 

T=o 
without  assuming  that  L^—R^  is  zero.     Consequently  (^^j  can  be 
satisfied  without  equation  (2)  being  satisfied  ;  that  is,  without  (c) 


Systems  of  Equations.  93 

being  satisfied.     Therefore  (d)  is  not  equivalent  to  (c)  but  to  the 
two  systems 

L-R=o 

L-R.=o 


L,-R„= 

I.-R=o] 
T=o  I 

U-R„=o] 

5.  Examples.  The  derivation  discussed  in  the  above  theorem 
is  the  one  so  frequently  used  in  elimination.  Thus  take  the 
system 

2.r-hj'=i7         (y)\ 
^x—\oy=   5         (2)  S 
Multiply  (\)  through  by  5  and  (2)  through  by  2  and  obtain  a 
new  equation  b}^  subtracting  the  former  from  the  latter  and  the 
system  becomes 

2.r+r=i7         (7i)} 

We  have  eliminated  x  from  the  second  equation  and  consequently 
y  is  readily  found  to  equal  3. 

From  (■})),  X  is  then  found  to  equal  7. 

The  theorem  shows  it  is  also  legitimate  to  transfonn 

into  ^     '  o    .    , 

6— 3.r=o  j 

by  multiplying  the  first  equation  through  by  x  and  subtracting 

the  resulting  equation  from  the  second. 

An  example  of  the  use  of  the  following  theorem  will  be  found 

in  V,  Art.  12  (c). 

6,  Theorem.     It  is  legitimate  to  derive  from  the  system 

the  system 

SL/+TL  =SR-"-f  TR.  \  '  ^^ 


//"  T  is  a  known  quantity,  not  zero. 


94 


Algebra. 


L -R  =o  (I)  I 

Iv  -R  =0  (2)    \  ^     ^ 


Rewrite  (a)  and  (d)  so  that  they  shall  read 

L,-R=o  0 

Iv-R=o  (: 

and  L-R,  =  o         (:,)  / 

s(i.r-R,o+T(L.-Rj=o     r4;  ^'       ^^ 

It  is  evident  that  any  set  of  values  which  will  satisfy  (^)  will 
satisfy  (d),  for  whatever  makes  ly,  — R^  and  L^— R„  each  zero  will 
satisfy  (d). 

It  is  seen  from  (t,)  that  any  set  of  values  that  satisfies  (d)  must 
make  ly,— R,  zero.     Equation  (4.)  will  then  become 

Ta.-R,)=o.  (5) 

Now,  since  T  is  a  known  quantity  not  zero,  this  cannot  be  sat- 
isfied unless  L^— R^  is  zero.  Hence  any  set  of  values  that  satisfies 
(d)  must  make  L,— R,  and  L,— R,  each  zero  ;  that  is,  must  be  a 
solution  of  (c). 

Now,  since  any  solution  of  (r)  is  a  solution  of  (d)  and  any 
solution  of  (d)  is  a  solution  of  (r),  the  two  systems  are  equivalent. 

7.  Theorem.  //'  from  a  system  containing  tivo  unknowji 
quantities 


zve  derive  the  systetn 


L=R,        (2)  \  ("^ 

L=R.  (z)l 


the  derivation  is  questionable  if  ly^  and  R^  both  involve  unknoivn 
quantities,  but  legitimate  if  either  is  a  kyiozvn  quantity  7iot  zero. 

First,  suppose  that  L,  and  R^  each  involve  unknown  quantities. 

Any  value  of  the  unknown  quantities  which  will  satisfy  the 
equation 

L=o 
must  satisfy  equation  (^),  since  the  relation  L,=Ri  must  hold  if 
system  (b)  is  to  be  satisfied. 

Also  any  value  of  the  unknown  quantities  which  will  satisfy 
the  equation 

R=o 
must   satisfy    (\),    since   the   relation    L,=R^  must  hold  if  the 
system  is  to  be  satisfied. 


Systems  of  Equations.  95 

Moreover,  any  value  of  the  unknown  quantities  which  will 
satisfv 

L,=  R, 
mitst  satisfy  (4.)  since  the  relation  L,  =  R,  must  hold. 

Therefore,  from  these  considerations,  it  is  evident  that  the 
system  (d)  is  not  equivalent  to  system  (a),  but  to  the  three 
systems 

Second,  suppose  that  either  ly^  or  R^  is  a  known  quantity  not 
zero. 

One  of  them,  say  R^,  is  the  known  quantity.  Therefore  L^ 
cannot  be  zero,  since  the  relation  Lj=Rj  must  hold.  Hence  the 
introduced  system  (bj  and  (bj  are  absurdities,  since  they  require 
that  Iv^  and  R^  be  zero.  Consequently  the  derivation  is  legitimate 
since  the  introduced  systems  are  incompatible. 

8.  Examples.  As  an  illustration  of  the  theorem,  consider  the 
sj^stem 

.    •  jf— 4=6— r  / 

2.r+i'=i3       \ 
This  is  satisfied  by  .1  =  3  and  j'=7.     Now  form  the  system 

.V— 4=6— J^'  ) 

which  is  satisfied  by  either  of  the  sets  of  values,  .1  =  3,  )'=7  or 
.r=4,  j/=6.  The  additional  solution  may  be  obtained  from  either 
of  the  systems 

.1  —  4=6—1' ) 

.1—4=0        ) 

.V— 4=6— r ) 

6— r=o        \ 
As  another  example  consider  the  system 

,r-f  2j'=7  j 


96  Algebra. 

which  is  satisfied  by  -v=5,  y=i.  From  this  we  may  obtain  the 
system 

X—2J'=2>     } 

From  the  first  equation  of  the  system 

Substituting  this  value  for  x  in  the  second  equation,  it  becomes 

9  + 1 2v + 4.y — 4 J'"  =  2 1  ; 
whence  j'=  i . 

Therefore,  from  the  first  equation  of  the  system, 

x=s. 
In  this  case  we  see  that  no  solution  has  been  introduced.     In 
fact,  the  introduced  systems  become 

X— 2r=3| 

A'— 2JI'=0  j 
X—2J=2>\ 

7  =  oj 
which  are  incompatible. 

9,  Theorem.     If /mm  the  system 
we  derive  the  system 

lCr,;}  ^^^ 

the  derivatio7i  is  questionaale  if  both  L^  ayid  R^  involve  tuiknoivn 
quantities,  but  legitimate  if  either  is  a  known  quantity  7iot  zero. 

First,  suppose  that  L,  and  R^  both  involve  unknown  quantities. 

Then,  by  Art.  7,  if  we  pass  from  (b)  to  (a)  we  gain  solutions. 
Hence  to  pass  from  (a)  to  (b)  is  to  lose  those  solutions. 

Second,  suppose  that  either  L,  or  R^  is  a  known  quantity. 

Then,  by  Art.  7,  if  we  pass  from  (b)  to  (a)  no  solutions  are 
gained.     Hence  none  are  lost  if  we  pass  from  (a)  to  (b). 

10.  Examples.  According  to  the  above  theorem  it  is  legiti- 
mate to  divide  one  equation  by  another,  member  by  member,  if 
one  member  is  a  known  quantity  not  zero. 


Systems  of  Equations.  97 

Thus  take  the  system 


and  derive  the  system 


The  only  set  of  values  which  will  satisfy  (a)  is  ji:=4,  jk=i. 
This  set  satisfies  (b)  and  no  solution  is  lost. 

The  system  (a)  is  equivalent  to  the  system  (b)  and  to  two 
other  systems  (see  Art.  7),  but  the  other  two  systems  are  incom- 
patible. 

As  an  example  of  the  case  in  which  solutions  may  be  lost, 
consider  the  system 


which  is  satisfied  by  either  of  the  two  sets  Jt:=o,  j/=3  and  ^==3, 
jj/=o.  If  we  divide  the  second  equation  by  the  first,  member  by 
member,  we  pass  to  the  system 

x=2,-y 


which  is  satisfied  only  by  the  values  ^=3,  y=o. 

II.  Solution  of  a  Linrar-Quadratic  System.  We  now 
propose  to  take  up  the  solution  of  those  systems  involving  two 
unknowns  which  consist  of  one  linear  and  one  quadratic  equation. 
It  is  convenient  to  call  this  a  linear-quadratic  system.  We  will 
proceed  by  first  working  the  following  particular  example  : 

jr=— 2y=i  (2))  '^ 

From  equation  ( i )  the  value  of  x  in  terms  of  j  is  easily  seen  to  be 

Substituting  this  value  of  .r  in  equation  (2)  we  obtain 

or  25— loj'+y— 2^=1  (^) 

Unitijig  and  transposing  terms 

y+ioj=24,  .     (6) 

whence,  solving  this  quadratic, 

_>'=2  or  —12, 
and  from  equation  (\)  ' 

.r=3ori7. 

A— 12 


98  Algebra. 

Consequently  there  are  hvo  sets  of  values  which  will  satisfy 
system  (a),  namely, 

and  x=i7,  )/=  — 12. 

Now  the  method  here  used  may  be  applied  to  the  solution  of 
any  linear-quadratic  system  containing  two  unknowns.     In  fact, 

take  the  general  case* 

x-\-ay=b  \  .^. 

x''-{-c}r-\-dxy-^ex-\-fy=g  \  ^ 

where  a,  b,  c,  d,  e,  f,  and  g  are  supposed  to  stand  ior  any  real 
quantities  whatever. 

The  value  of  x  in  terms  of  y  from  the  first  equation  of  the 

system  is 

x=b—ay  (-J) 

Substituting  this  for  x  in  the  second  equation  of  the  system,  that 
equation  becomes 

b"—2aby-\-a'y^'-\-CT^bdy—ady~-\-eb—aey^fy^=g. 
Combining  together  those  terms  which  contain  jk^  and  those 
which  contain  y  and  transposing  all  the  known  terms  to  the  right 
hand  side  of  the  equation,  this  becomes 

(a"  -f  c—ad)y''-^  (bd—  2ab—ae-\-f)y=g—  b'—eb, 
which  is  a  quadratic  in  which  y  is  the  only  unknown  quantity, 
whence  it  can  be  solved.     The  values  which  may  be  found  from 
this  can  be  substituted  in  equation  ('])  above,  and  the  values  of 
.V  will  be  determined. 

12.  Examples.     Solve  the  following  systems  : 

J        f      x4-j'=7 
(a-^ -I- 21/^=74. 

\2x-i-xy-\-2y=^i6. 

I  -r-f  r=4 
^*       I       xy=g6. 

*It  might  be  thought  that  this  is  not  a  general  case,  since  x  in  the  first  equation  aufl 
x^in  the  second  do  not  appear  with  coefficients.  But  if  either  of  them  had  a  coefficient 
the  equation  could  be  reduced  to  the  given  form  by  dividing  through  by  that  very 
coefficient. 


Systems  of  Equations.  99 

r        -v— r 
■>-     ,-=4 

\^x     .v—v 

13.  Solution  of  Systems  of  two  Quadratics.  If  we  have 
a  system  of  two  quadratic  equations  containing  two  unknown 
quantities  and  attempt  to  eliminate  one  of  the  unknown  qiyintities 
it  will  be  found  in  general  that  the  resulting  equation  is  of  the 
fourth  degree.     Thus  take  the  system 

y'-^xy—\o.  1 
We  find  from  the  first  equation  that 

SubvStitute  this  value  for  y  in  the  vSecond  equation,  and  it  be- 
comes 

or,  expanding  and  collecting  terms, 

.^^-f.r^ — 5JI* — 5J' — 25=  10. 
Now,  since  we  are  not  yet  familiar  with  the  solution  of  equa- 
tions of  a  degree  higher  than  the  second,  the  treatment  of  sys- 
tems of  two  quadratics  in  general  cannot  be  taken  up  at  this 
place.  But  there  are  two  important  special  cases  of  svstems  of 
two  quadratics  whose  treatment  will  involve  no  knowledge  be- 
yond the  solution  of  quadratic  equations,  and  these  we  will  now 
consider.     The  cases  referred  to  are     \ 

I.  Where  the  terms  in  each  equation  containing  the  un- 
known quantities  constitute  a  homogeneous  expression  with  respect 
to  the  unknown  quantities. 

II.  Where  the  equations  are  symmetrical.* 

14.  Case  I.  We  will  illustrate  the  first  case,  and  also  the 
method  of  elimination  which  may  be  applied  to  any  example  of 
it,  by  the  following  solution  : 

Solve  the  svstem  ^-      ..         -^     ^^  >    \ 

3-V'— ioy=35.  (2) 


♦For  the  deflnitione  of  boniogeueous  and  symmetrical  see  I,  Arts.  7  and  8. 


lOO  A1.GEBRA. 

Suppose  x=z>j',  where  z'  is  a  new  unknown  quantity.     Then, 
substituting  this  in  the  equations,  the  system  becomes 

From  equation  (2,)  we  find  that 


and  from  equation  (4:) 


(a) 


Whence,  from  (5J  and  (b) 

5  35 


r=,-75^  re; 


V^  —  2V       327'— 10. 

Clearing  of  fractions, 

15-^^— 50=352'=— 7oz\ 
Transposing  and  uniting  terms,  and  dividing  through  by  lo 

2z;=-7Z;=-5. 

Solving  this  quadratic 

z'=|-  or  T. 
Substituting  the  first  of  these  values  in  (^3J  we  obtoin 

Whence-  jj/  =  d=  2 

and  since  x^=vy 

Now,  substituting  the  second  value  of  v  in  (2,)  we  obtain 

y_2r==5. 

Whence  j'=  db  V  —  5 

and,  since  :t-=z_;;/, 

x=±\/— 5- 
Therefore  we  have,  as  the  solution   of  the  original  syvStem,  the 
four  sets  of  values. 

x=5,  y=2\     x=  — 5,    jj'=  — 2;     x=\/  — 5,   j'=V  — 5; 

A-=  — V— 5,  j/=  — V— 5- 
The  general  system  of  two|equations  of  the  above  class  may  be 
represented  by 

x"  +  axy  -\r  byr^c  \  , ,  > 

x'^-dxy^cy-'^^f.]  '    -^ 


(F.    &^h-^ 


Systems  of  Equations. 


lOI 


The  student  may  show  that  the  method  set  forth  above  will 
solve  the  general  system  and  hence  any  possible  example  under  it. 


15. 

Examples.     Solve  the  following  systems  : 

J  ^ 

f  ^+^=3* 

2. 

t 

Jf      X        - 

I    xy         '■- 

3- 

{x^+xy=is 
\xy-f=2. 

4-' 

\        xy=ii. 

^x-\-y  ^x-y     lo 

5- 

^x-y     x-\-y      3 

( 

x'+y=45. 

16-  Case  II.  To  show  that  any  system  of  two  sj^mmetrical 
quadratics  can  be  solved,  we  will  start  with  the  general  case, 
which  is  evidently. 

x'-j-ax-\-dxy-\-ay-4-y^=c  \  ,    . 

x'-\-dx+exy-\-dy+y'=/\  ^^^ 

If  x^  and  jj/^  appeared  in  either  of  these  equations  with  coeffic- 
ients the  system  could  be  reduced  to  the  given  form  by  dividing 
the  equation  through  by  that  coefficient. 

Through  the  giv^en  system  substitute  21 -{-w  for  x  and  21— w  for 
y,  where  ti  and  w  are  two  new  unknown  quantities.  Then  (a) 
becomes 

22i^-\-2Zi^-\-2au-{-b2i''—bz(f=^c\  ,,v 

22r-\-2u<^-\-2dii-\-eic' — ew'^f )  ^    ^ 

Subtracting  the  second  of  these  equations  from  the  first  we  obtain 

2(a—d)2i-\-(b—e)ie—(b—e)ur=c—f, 
or  2(a—d)u-\-(b—e)(u'—'iif)=c—f. 


Whence 


c—f—(b—e)(u'—iif) 


2(a—d) 

Now  if  the  right-hand  side  of  this  equation  be  substituted  for 
21  in  the  terms  2a2c  and  2d2i  of  system  (b),  that  system  will  con- 
tain no  powers  of  the  unknown  quantities  but  the  second  and  will 


I02  Algkbra. 

therefore  come  under  Case  I.     When  //  and  ?»'  are  thus  found,   x 
and  J'  can  be  detennined  from  the  equations. 

The  above  work  shows  that  Case  II  can  ahva^-s  be  solved,  but 
we  do  not  pretend  that  the  method  used  is  always  the  most  eco- 
nomical one  to  employ.  The  insight  and  ingenuity  of  the  student 
will  often  suggest  special  expedients  for  particular  examples 
which  are  prefera])le  to  a  general  method. 

17.  KxAMPLKS.     Solve  the  following  systems: 

^        (  x-\-xy-\-y=6s 
\         xy        =50. 

Ur=  -f-  X  -\-y  4-J'^  =  2  6 . 

1^+^  =  14 
S-       I  X    y     45 

^'      I       .i:r=6. 
A  common  expedient  for  readily  j^oh-ing  such   a  system  is  to 
first  transform  it  into  the  svstem 


1  x^-  —  2xy-\-y'=^6\ 

i  A"+2.i:r-fj'-'=3oJ- 

from  which  the  values  of  x—y  and  x-\-y  can  be  found  and  conse 

quently  the  values  of  x  and  y. 

:^±fj.^ 

_ 

x-\-y        9 

^  • 

-i-+j'_i8 

'"^y  ~ii 

18.  Miscellaneous  Systems.  We  have  enumerated  all  the 
classes  of  systems  involving  equations  of  a  degree  higher  than 
the  first  which  can  invariably  be  solved  without  a  knowledge  of 
the  solution  of  cubic  and  higher  equations.  The  solvable  cases 
embrace  but  a  small  fraction  of  the  S3'Stems  which  ma}^  arise.  Of 
the  numbers  remaining  a  still  smaller  proportion  can  be  solved  by 
special  expedients.  The  great  mass  of  systems  involving  quad- 
ratic or  higher  equations  are  thus  irreducible  by  straightforward 
methods  of  solution,  /.  e.,  as  a  rule,  such  systems  are  insolvable,  not- 


Systems  of  Equations.  103 

withstanding  a  chance  exception.  In  those  systems  which  may 
be  solved,  special  expedients  are  more  to  be  sought  for  than  gen- 
eral methods.  In  fact,  sharp  inspection  of  the  equations  and  a 
knowledge  of  algebraic  forms  will  often  be  the  means  of  discover- 
ing an  ingenious  solution  for  an  apparently  insolvable  system. 

The  theorems  of  this  chapter  will  be  found  useful  either  in  just- 
ifying or  in  throwing  doubt  upon  many  of  the  common  transfor- 
mations during  the  ordinary  solution  of  a  system.  As  far  as 
possible  the  student  should  endeavor  to  take  account  of  all  ques- 
tionable derivations  at  the  time  they  are  made  and  make  allow- 
ance for  them  in  the  result.  In  this  connection  the  remarks  at 
the  beginning  of  the  last  chapter  should  not  be  forgotten.  No 
matter  how  skillfully  or  ingeniou.sly  a  set  of  values  may  have 
been  obtained,  they  must  satisfy  the  original  system,  ot  it  is  no  solu- 
tion. Whenever  am-  derivation  not  allowed  by  the  theorems  is 
used,  however  plausible  it  may  seem,  this  ultimate  test  must  be 
applied. 

The  following  systems  include  examples  of  the  cases  already 
considered,  besides  others  requiring  special  treatment.  The 
method  of  Case  II  will  be  found  to  solve  many  symmetrical  sys- 
tems of  high  degree. 

.       !    -r  +  r=5  rO 

•       i.v5-fjp=275.  (2) 

Put  .r  =//-frt'  and  r=//  — zf,  whence  the  system  becomes 

\  2H^-\-20U\i^-\-  10^71^=275. 

From  the  first  of  these  equations  ?^=f .  Whence,  substituting 
this  in  the  second,  we  obtain 

^125  ,  62s    .  ,  12s   ^ 
16         2  2 

which  is  a  quadratic  in  terms  of  w\     When  iv  is  found  x  and    r 
can  be  found  from  the  equations  x=u-\-iu  and  )'=7/-— rr. 
(    X- 1  >^  ==  I  ox\ '  -f  9000 
I  .r--hj'=200. 

->       I    ■v-J'=2 
^'       |.v^— y=8. 

,        I    -^-fj'=5 
+  ■      ].r3  +  r^=65. 


I04 


Al^GKBRA. 


5-. 
6. 

7- 
8. 

9- 

lo. 

II. 

12. 

14. 

15' 

16. 

I?' 
18. 

19. 
20. 
21. 
22. 


t       xy=b. 

(  x^-\-xy-\-2y^=2 

{2x^-\-xy-\-  y=2. 

\x^—  xy-\-y=2i. 

Cxy+yx=2o 

V,  X    y     4. 

f  (x-\-y)xy=24.o 
\        x^-\-y^=28o. 
j  x^-hr-\-x+y=iS 
j  x-—y^-\-x—y=^6. 

\  (^-4J(y—7)=o. 

j  x^-\-y^—i2=x-\-y 
I  xy-i-8=2(x+y) 

i  jr*— Jt'^+y— j/^=84 
l9(y-x)=is 

I       x^—y''=(f, 

(   .r^ — xy=2 
(  2Jt:"+y=9 

jf  -  4-jF^ — -^  — j^' =  7  8 
jrK+x4-jF=30. 

f        .r=4-y=34 
(2J«;^— 3:1:^=23— 2^. 

^  V-^+Vj^=3 

f      sj  X—  sf y=-2 
\  (x-\-y)^^=^jo^ 


Result,  X- 


Result,  jr=4  or  |. 


y=2  or 


'    ;^=±   ^ 


V3  ^>/3^ 

Result,  ^-=9  or  — ^/+JV^39- 

J^/=3  or  — '/-• i-V-39. 

Result,  j»;==b5,  or  d=3. 

jj/==b3,  or  ±5. 

Result,  x=^. 

y=i. 

Result,  x=^,  or— 3. 
y=S,  or-5. 


Systems  of  Equations.  105 

r  xy=i22^  Result,  ^=49  or  25. 

lV-^+Vj=i2.  j=25or49. 

^x-\-y^a—s/  x-\-y 
I  x—y=  b. 

Result,  ^=^r^±A^+i^yV±l 
4    


24^ 


y 


A— 13 


CHAPTER  VIII. 

PROGRESSIONS. 

1.  Definitions.  An  Arithmetical  Progression  is  a  series  of 
terms  such  that  each  differs  from  the  preceding  by  a  fixed  quan- 
tity, called  the  common  differeiice.     The  following  are  examples  : 

74-9+11  +  13+15+  •    •    . 
31  +  26  +  214-16+11+  .    .    . 
a-^r(a-\-d)^(a^2d)^(a^-^d)^  ... 
(x—y)-\-x-^(x-\-y)-\-  .    .    . 
(jc-2,y)-^(x-y)^(x^y)-^(x^iy)-V  .    .    . 
The  first  ana  last  terms  of  any  given  progression  are  called  the 
Extremes,  and  the  other  terms  the  Means. 

2.  To  Find  the  «th  Term  of  an  Arithmetical  Pro- 
ORESSION.  Represent  the  first  term  of  the  progression  by  a  and 
the  common  differeiice  by  d.     Then  we  have 

Number  of  ter  711.  i.  2.  3.  4.  5. 

Progressio7i.  a  -\-  (a-\-d)  -\-  (a-\-2d)  +  (a-\-2fd)  +  (a-\-^d),  etc. 
We  notice  that  by  the  nature  of  the  progression  every  time  the 
number  of  terms  is  increased  by  i  the  coefficient  of  d  is  increased 
\>y  I  also  ;  hence  to  get  the  71  th  term  from  the  5th  term,  the  com- 
mon difference  must  be  added  to  it  n  —  ^  times.  Whence, 
representing  the  ;^th  term  by  /,  l=a-\-/^d-\-(7i  —  ^)d,  or 

l=a-\-(7i-i)d.  (i) 

3.  To  FIND  THE  Sum  of  n  Terms  of  an  Arithmetical. 
Progression.  Representing  the  sum  of  the  arithmetical  pro- 
gression by  s,  we  have 

s=a-i-(a  +  d)-\-(a  +  2d)-\-(a-h2,d)+  ...+/,  (i) 

or,  writing  this  progression  in  revense  order,  we  have 

s=l-{-(l-d)-^(l-2d)  +  (/-sd)-h  .    .    .  -\-a  (2) 

Now  adding  (i)  and  (2)  together  term  for  term,  noticing  that 
the  common  difference  vanishes,  we  have 

25=r«+/;+r^+/;+ra+/;+r«+/;+  .  .  .  +(a+i). 


Progressions.  107 

If  the  number  of  temis  in  the  original  progression  be  called  n^ 
this  becomes 

2S=^n(a-\-l)^ 

whence  s=-\n(a-]rl). 

4.  To  Insert  any  Number  of  Arithmetical  Means  be- 
tween TWO  GIVEN  Quantities.  Suppose  we  are  to  insert  p 
arithmetical  means  between  the  two  terms  a  and  /.  The  whole 
number  of  terms  in  the  progression  consists  of  the  r  means  and 
the  two  extremes.  Hence  the  number  of  tenns  in  the  progression 
is/>  +  2.     Therefore,  substituting  in  (\),  Art.  2,  we  obtain 

/=^  +  (^^-f  2— iX 
-     / —  a 

and  now,  since  the  common  difference  is  known,  any  number  of 
means  can  be  found  by  repeated  additions. 

5.  The  two  equations 
l^a-\-(n—\)d  ■  '     (1) 
s^\n(a-^l)  (2) 

contain  five  different  quantities.  If  any  two  of  them  are  unknown 
and  the  values  of  the  re5t  are  given  the  values  of  the  two  un- 
known can  be  determined  by  a  solution  of  the  system.  As  an 
example,  suppose  that  a  and  d  are  unknown  and  the  rest  known. 
Putting  X  for  a  and  y  for  d  so  that  the  unknown  quantities  will 
appear  in  their  usual  form,  the  system  becomes 

l=x+(n-i)y  (z) 

s=\n(x-^l)  (^) 

Finding  the  value  of  x  in  each  equation  the  sj^stem  becomes 

^x=l-(n-Y)y  (s) 

|.=  --/,  (6) 

Whence,  equating  right-hand  members  of  ($)  and  (b),  we  obtain 

/-r«-ii>'="-/, 

2?ll—2S  .     . 

whence  y—  -> .  (']) 

n{n—i) 


io8 


AlvGEBRA. 


Therefore,  restoring  a  and  dva  (6)  and  (-j), 

2S 

«= /, 

71 

-       2nl—2S 

d=— r. 

n{n — i) 
where  a  and  d  are  expressed  in  temis  of  the  three  known  quan- 
tities, n,  /,  snd  s.  In  Hke  manner  we  may  suppose  a7iy  two  of  the 
five  quantities  unknown  and  find  their  values  in  terms  of  the 
known  ones.  In  all,  ten  different  cases  may  arise,  which  are 
given  in  the  following  table,  each  of  which  is  to  be  worked  by 
the  student. 


No 

Given. 

I. 

a,  d,  n, 

2. 

/,   d,    71, 

3- 

a,  /,  71, 

4- 

a,  n,  s, 

5- 

71,  d,  s, 

6. 

/,    71,    s, 

7- 

a,  d,  /, 

8. 

a,  /,  s, 

9- 

a,  d,  s, 

lO. 

/,  d,  s, 

quired 


/,  s, 
a,  s, 

d,  s, 
d,  /, 
a,  /, 
a,d, 


l=a-i-(?i—i)d ; 

a=/—(?i—i)d; 

.     I—  a 
«= : 


d= 


71— \ 
2S—2a7l 


a= 


71(71-1)' 

S       {7l—l)d 


2S       J 
71 


I— a  , 
71,  s,  '71=—^ — hi ; 


Result. 


s=\n[2a-\-  {71—  i)d\ 
s^=\n[2l—  {72  —  i)d~\. 


d= 


I,  71,   /  =  —^d^s/2ds-\-ia—y.r:n-- 


a, 71,  a  =  \d±is/{l^\dY—ld^ 


7i{n—i)' 
{l-a){l-a-{-d) 

2d 

I'— a" 
2S  —  a—V 

d—2adt:^(2a—d)^Sd8 
2d 

2/+  dzt  x/W-fclf-^^Sdi^ 

2d 


6.  Examples. 

1.  Find  the  sum  of  g  tenns  of  the  progsession  3  +  7  +  1 1 ,  etc. 

2.  The  first  term  is  96,  the  common  difference  —5  ;  what  is 
the  1 3  th  term  ? 


Progressions.  i  09 

J.  The  first  term  is  8^,  the  common  difference  —  f,  and  the 
number  of  terms  29  ;  what  is  the  sum  ? 

4.  The  first  term  is  f,  the  common  difference  f,  and  the 
number  of  terms  1 2  ;  what  is  the  sum  ? 

5.  Insert  10  arithmetical  maeans  between  —J  and  -f  J. 

6.  Find  the  sum  of  the  first  n  odd  numbers  1+3  +  5  +  7,  ^^c. 

7.  Find  the  sum  of  7i  terms  of  the  progression  of  natural 
numbers  1  +  2  +  3  +  4,  ^tc. 

S.  Find  the  sum  of  fi  terms  of  the  progression  of  even 
numbers  0  +  2  +  4  +  6  +  8,  etc. 

p.  The  first  term  is  11,  the  common  difference  —2,  ^nd  the 
sum  27.     Find  the  number  of  terms. 

10.  The  first  term  is  4,  the  common  difference  is  2,  and  the 
sum  18.     Find  the  number  of  terms. 

11.  The  first  term  is  11,  the  common  difference  is  —3,  and 
the  sum  24.     Find  the  number  of  terms. 

12.  The  sum  of  ?t  consecutive  odd  numbers  is  s.  Find  the 
first  of  the  numbers. 

ij.  Select  10  consecutive  numbers  from  the  natural  scale 
whose  sum  shall  be  1000. 

14.     Sum  Vi+ V2  +  3Vi+  etc.,  to  twenty  terms. 

zy.     Sum  5  —  2  —  9—16—  etc.,  to  eight  terms. 

16.  Find  the  tenth  term  of  the  arithmetical  progression 
whose  first  and  sixteenth  terms  are  2  and  48  ;  and  also  detenn- 
ine  the  sum  of  those  eight  terms  the  last  of  which  is  60. 

77.     Insert  five  arithmetical  means  between  10  and  8. 

iS.     Insert  four  arithmetical  means  between  —2  and  —16. 

ip.  How  many  terms  must  be  taken  from  the  commence- 
ment of  the  series  1  +  5  +  9+13+17  etc. ,  so  that  the  sum  of  the  1 3 
succeeding  terms  may  be  741  ? 

20.  Wnat  is  the  expression  for  the  sum  of  ;/  terms  of  an 
arithmetical  progression  whose  first  term  is  f  and  the  difference 
of  whose  third  and  seventh  terms  is  3  ? 

21.  The  sum  of  the  first  three  terms  of  an  arithmetical  pro- 


no  Algebra. 

gression  is  15  and  the  sum  of  their  squares  is  83  ;  find  the  com- 
mon difference. 

22.  There  are  two  arithmetrical  series  which  have  the  same 
common  difference  ;  the  first  terms  are  3  and  5  respectively  and 
the  sum  of  seven  terms  of  the  one  is  to  the  sum  of  seven  terms  of 
the  other  as  2  to  3.     Determine  the  series. 

7.  Definitions.  A  Geometrical  Progression  is  3.  s&ri^s  oi  t^rms 
such  that  each  is  the  product  of  the  preceding  by  a  fixed  factor 
called  the  Ratio.     The  following  are  examples  : 

3  +  6+12  +  24  +  48,  etc. 
100+50  +  25+ i2|-+6^,  etc. 
i  +  i+i+xV+sV,  etc. 
i+i  +  TVH-2V+4V  etc.^ 
The  first  and  last  terms  of  any  progression  are  often  called  the 
Extremes  and  the  remaining  terms  the  Means. 

8.  To  Find  the  n  th  Term.  I^et  a  represent  the  first  term 
of  the  geometrical  progression  and  r  the  ratio.  Then  the  pro- 
gression may  be  written : 

Nnmber  of  term.      i.    2.       3.       4.        5. 
Progressioji.  a  -\-ar-\-  ar^-\-  ar^  +  ar'^. 

We  notice  that,  by  the  nature  of  the  progression,  every  time 
the  number  of  terms  is  increased  i  the  exponent  of  r  is  increased 
by  I  also  ;  hence  to  get  the  ?i  th  term  from  the  5th  term  it  must 
be  multiplied  by  the  ratio  ^—5  times.  Whence,  reoresenting  the 
nth  term  by  /,  and  /=(ar^)(n—2,), 
or 

l=ar"-\  (i) 

9.  To  Find  the  Sum  of  ?^  Terms.  Represnting  the  sum  of 
the  geometrical  progression  by  ^  we  have 

s=^a-{-ar-\-ar^-^ar^-\-  .    .    .  -\-ar"~''-\-ar"~\        (i) 
Multiplying  this  equation  through  by  r—  1 ,  we  obtain 

(r — i)s=a?'" — a. 
Whence 

ar"—  a 


r—i 


(2) 


Now  ar"  —  r(ar"  ').     Therefore,  since  ar"  '=/, 

ar"=al. 


Progressions.  i  i  i 

Whence,  substituting  this  value  of  ar"  in  (2)  we  obtain  as  an- 
other expression  for  ^ 

rl—a 

s= (2,) 

10.  To  Insert  any  Number  of  Geometrical  Means  Be- 
tween TWO  Given  Quantities.  Suppose  we  are  to  insert  p 
geometrical  means  between  the  two  terms  a  and  /.  The  whole 
number  of  terms  in  the  progression  is  therefore  p-\-2.  Hence, 
substituting /-f^  for  7i  in  (1),  Art.  8, 

l=aa"-\ 
Consequently 

and  now,  since  the  ratio  is  known,  any  number  of  means  can  be 
found  by  repeated  multiplications. 

11.  The  two  equations  '    • 

I        ar" — a 

s= 

V,  r — I 

contain  five  different  quantities.  If  any  two  of  them  are  unknown, 
and  the  values  of  the  rest  are  given,  the  values  of  the  two  un- 
known can  be  determined  by  a  solution  of  the  system.  But  if  r 
is  an  unknown  quantit}^  the  equations  of  the  system  are  of  a  high 
degree,  since  n  is  usually  a  large  number  and  always  greater  than 
2  at  least.  In  this  case  we  will  be  unable  to  solve  the  system,  as 
it  is  one  beyond  the  range  cf  Chapter  VII.  Also  if  n  is  an  un- 
known quantity,  we  will  have  an  equation  with  the  unknown 
quantity  appearing  as  an  exponent,  which  is  a  kind  of  equation 
we  have  not  yet  discussed.  Hence  there  are  a  limited  number  of 
cases  in  which  we  can  solve  the  above  system.  The  following  ta- 
ble contains  the  ten  possible  cases,  with  the  solutioni  as  far  as 
possible.  The  values  of  n  in  the  last  four  are  printed  merely  to 
make  the  table  complete,  for  the  manner  of  obtaining  them  is  not 
explained  until  Chapter  XV  is  reached. 


112 


AlvGEBRA. 


No 


I. 


Given. 


'J  '  1  ""I 


2.  /,  r,  ;^, 

n,  r,  s, 


quire,  d 


Result, 


/,   s,   l=ar"-'] 

(r-i>_ 


a,  s,  «= 


a,  L   a= 


r,  5, 


/,     71,   S, 

a,  r,   / 
a.  L  s. 


1- 
8. 

9- 
lo.  /,   r,  5, 


<2.  r,  ^, 


5,  ;?, 
r,  71, 


r"— I    ' 


s= 


ar" —  a 
r — I 

la"— I 

{r—i)sr"-' 
r"~i 

l''^^—a^i 


s= 


_L  1 


ar^' — rs—a—s  ;    l(s—iy'  '=a(s—ay'  \ 
a(s—ay-'=l(s—iy-';     (s—l)r"—sr"-'=—l. 


Ir—a  _log  ^— log  a 
r— I  log  r 

^— a  _  log  /—log  « 

'5—/'  log  (^—«;— log  (^—/) 


+  1. 


/  ..     ,_^  +  (r-i>^  log  [^  +  (r-i)^]-log  g 

r  log  r 

log /— log  [/r— (r— 1)>?] 

a, 71,  a=lr—{r—\)s\  71=—^^ ^-^ ^ ^  +1. 

log  r 


12.  Examples  and  Problems. 

1.  Find   the   sum   of    10   terms   of    the   progression   3  +  9 
H-27H-etc. 

2.  Find  the  sum  of  10  terms  of  the  series  fV+xf  o^  +  tAtt* 
etc.,  or  the  series  .333  + 

j.     Find  the  sum  of  100  terms  of  the  progression  .3333+  etc. 

4.  Sum  5  terms  of  the  progression  27  +  270+2700,  etc. 

5.  Sum  10  terms  of  the  progression  4—2+1—  etc. 

6.  Sum  the  series  V3  +  >/6  +  vi2+  etc.,  to  eight  terms. 

7.  Sum  the  series  3  — 2  +  |— f  +  etc.,  to  nine  terms. 

8.  Sum  the  series  —4  +  8—16  +  32,  etc.,  to  6  terms. 

p.     The  fourth  term  of  a  geometrical  progression  is  192  and 
the  seventh  term  is  12288  ;  find  the  sum  of  the  first  three  terms. 


Progressions.  113 

10.  Prove  that  if  quantities  be  in  geometric  progression  their 
differences  are  also  in  geometrical  progression,  having  the  same 
common  ratio  as  before. 

11.  The  first  and  sixth  terms  of  a  geometric  progression  are 
I  and  243  ;  find  the  sum  of  six  terms,  commencing  at  the  third. 

12.  The  first  term  of  a  geometric  progression  is  5  and  the 
ratio  2.  How  many  terms  of  this  series  must  be  taken  that  their 
sum  may  be  equal  to  33  times  the  sum  of  half  as  many  terms  ? 

A— 14 


CHAPTER  IX. 

ARRANGEMENTS    AND    GROUPS. 

1.  Definitions.  Every  different  order  in  which  given  things 
can  be  placed  is  called  an  Arrangement  or  Pertmitation,  and  every 
different  selection  that  can  be  made  is  called  a  Group  or 
Combination. 

Thus  if  we  take  the  letters  a^  b,  c  two  at  a  time  there  are  six 
arrangements,  viz  : 

ab,  ac,  ba,  be,  ea,  eb, 

but  there  are  only  three  groups,  viz  : 

ab,  ae,  be. 

If  we  take  the  letters  a,  b,  c  all  at  a  time,  there  are  six 
arrangements,  viz  : 

abe,  acb,  bae,  bca,  cab,  cba, 
but  there  is  only  one  group,  viz  : 

abc. 

2..  probi.em.  to  find  the  number  of  arrangements  of 
71  Different  Things  taken  All  at  a  time. 

First.  If  we  take  one  thing,  say  the  letter  a,  there  can  be  but 
one  arrangement,  viz  :  the  thing  itself. 

Second.  If  we  take  two  things,  say  the  letters  a  and  b,  there 
are  two  arrangements,  viz  : 

ab,  ba. 

Third.  If  we  take  three  things,  say  a,  b,  r,  there  are  six 
arrangements,  viz  : 

abc,  acb,  bac,  bca,  cab,  cba. 

Notice  that  there  are  two  arrangements  in  which  a  stands  first, 
two  more  in  which  b  stands  first,  and  two  more  in  which  c  stands 
first. 

Fourth.  If  w^e  take  four  things,  say  «.  b,  c,  d,  then  we  may 
arrange  the  three  letters  b,'C,  d  in  every  possible  way  and  place  a 
before  each  arrangement,  then  arrange  the  three  letters  a,  c,  d  in 
every  possible  way  and  place  b  before  each  arrangement,  then 
arrange  the  three  letters  a,  b,  d  in  every  possible  way  and  place 


Arrangements  and  Groups.  115 

the  letter  c  before  each  arrangement,  and  finally  arrange  the  three 
letters  a,  b,  r  in  ev^ery  possible  way  and  place  the  letter  d  before 
each  arrangement.  It  is  evident  that  all  four  letters  a,  b,  c,  d 
appear  in  each  arrangement  thus  formed,  and  it  is  also  evident 
that  the  number  of  arrangements  in  which  a  stands  first  is  exactly 
the  same  as  the  number  in  which  b  stands  first,  and  so  on. 

Hence  there  are  in  all  four  times  as  many  arrangements  of  four 
things  taking  all  at  a  time  as  there  are  of  three  things  taking  all 
at  a  time,  or  there  are  four  times  six  or  twenty-four  arrangements 
of  four  things  taking  all  at  a  time. 

hi  geyieral,  if  we  have  n  things,  say  the  letters  a,  b,  r,  d,  e,f,  .  . 
then  we  may  suppose  all  the  letters  but  a  arranged  in  every  pos- 
sible order  and  then  a  placed  before  each  of  these  arrangements  ; 
then  we  may  suppose  all  the  letters  but  b  arranged  in  every  pos- 
sible order  and  then  b  placed  before  each  of  thCvSe  arrangements, 
and  so  on. 

It  is  evident  that  all  71  letters  appear  in  each  arrangement  thus 
formed,  and  it  is  also  evident  that  the  number  of  arrangements  in 
which  a  stands  first  is  exactly  the  same  as  the  number  in  which 
any  other  letter  stands  first. 

Now  the  number  of  arrangements  in  which  a  stands  first  is  evi- 
dently the  number  of  arrangements  of  (n—i)  things  taken  all  at 
a  time,  and  hence  the  total  number  of  arrangements  of  n  things 
taking  all  at  a  time  is  w  times  the  number  of  arrangements  of  ;^—  i 
things  taking  all  at  a  time. 

Let  us  represent  the  number  of  arrangements  of  ?i  things  taking 
all  at  a  time  b}^  A„  and  the  number  of  arrangements  of  ?i—i 
things  taken  all  a  time  by  A;,_i,  etc.  Then  by  what  has  just 
been  shown  we  have 

A„_2  =  (?i-2)A,,_^, 

A3  =  3A2, 

A2^2Ai , 


Il6  Al^GEBRA. 

Now  multiply  these  equations  together,  member  b}^  member, 
and  we  get 

Aj  Ao  A3   .   .   .  A„=2Ai  3A2   .   .   .   ??A„_i 

=  1x2x3  .   .   .   ;/ Ai  A.2   .   .   .  A„_i. 
By  cancelling  common  factors  we  get 

A;,=  i  X  2x3    .    .    .    71. 

The  product  of  the  integer  numbers  from  71  down  to  i  or  from 
I  up  to  71  is  often  represented  by  j^^  or  ;?!  ,  and  is  read  factorial  w, 
or  71  admiration. 

With  this  notation  we  may  write 

3.  ProbIvEm.  To  find  the  Number  of  Arrangements  of 
?^  Things  taken  r  at  a  Time. 

ivCt  us  first  take  a  particular  case,  say  the  number  of  arrange- 
ments of  five  things,  say  the  five  letters  a,  b,  c,  d,  e,  taken  three 
at  a  time.  .  Suppose  the  arrangements  all  made  and  we  select 
those  which  begin  with  a  and  put  them  by  themselves  in  one 
class,  then  those  which  begin  with  b  and  put  them  by  themselves 
in  another  class,  and  so  on.  We  then  divide  the  whole  number 
of  arrangements  into  five  classes,  and  it  is  evident  that  the  num- 
ber in  any  one  class  is  just  the  same  as  in  any  other  class. 
Consider  those  which  begin  with  a.  Then  every  arrangement  in 
this  class  contains  besides  a  two  of  the  four  letters  b,  c,  d,  e,  and 
since  a  is  fixed  and  the  other  letters  arranged  in  every  possible 
way,  therefore  the  number  of  these  arrangements  must  equal 
the  number  of  arrangements  of  the  four  letters  b,  c,  d,  e  taken 
two  at  a  time. 

Iti  g€7ieral,  if  we  have  7z  things,  say  the  letters  a,  b,  c,  d,  e,f,  .  . 
to  be  taken  r  at  a  time,  we  may  select  all  those  arrangements 
which  begin  with  a  and  put  them  by  themselves  in  one  class,  then 
those  which  begin  with  b  and  put  them  by  themselves  in  another 
class,  and  so  on.  We  thus  divide  the  whole  number  of  arrange- 
ments into  71  classes,  and  it  is  evident  that  the  number  of 
arrangements  in  any  one  class  is  just  the  same  as  the  number  of 
arrangements  in  any  other  class. 

Consider  those  which  begin  with  a. 


Arrangements  and  Groups.  117 

Then  every  arrangement  in  this  class  contains  besides  a,  (r—i) 
of  the  letters  b,  c,  d,  .  .  ,  and  since  a  is  fixed  while  the  remain- 
ing letters  are  arranged  in  every  possible  order,  therefore  the  number 
of  arrangements  in  the  class  considered  must  equal  the  number  of 
arrangements  of  71  — \  letters  b,  c,  d,  .    .  ,  taken  r— i  at  a  time. 

As  there  are  71  such  classes  and  as  the  number  of  arrangements 
in  each  class  equals  the  number  of  arrangements  of  n—\  things 
taking  r—  i  at  a  time,  therefore  the  total  number  of  arrangements  of 
n  things  taken  r  at  a  time  equals  ;2  times  the  number  of  arrange- 
ments of  ?^—  I  things  taken  r— i  at  a  time. 

Let  us  represent  the  number  of  arrangements  of  ?i  things  taken 
r  at  a  time  by  A(")  and  similarily  any  number  of  things  taken 
any  number  at  a  time,  say  s  things  taken  /  at  a  time  (s  being 
greater  than  0  by  A(;),  then  by  what  has  just  been  proved 

A(';)  =  nAOz\) 
A(':rl)  =  (n-i)A(':r?:) 

A('{-'"^')  =  (n-r+i). 
Multipl}^  these  equations  together,    member  by  member,  and 
cancel  common  factors  and  we  get 

AC'.)  =  7l(?l-j)(7l-2)   .     .     .  (7i-r+i). 
Multiply    and    then     divide    the     right-hand    member     by 
(n  —  r)(7i  —  r-{-i)  .    .    .  i  and  we  get 

(7f~r)(7i  —  r—i  ...  I 
It  is  easily  .seen  that  the  numerator  is  1 71  and  the  denominator 
is  \n—r,  hence 

A(';)=,'- 

71  —  7' 

4.  PROBI.KM.  To  FIND  THE  NUMBER  OF  GROUPS  OF  ;/  DIF- 
FERENT Things  taken  r  at  a  Time. 

.    Take  the  letters  a,  b,  c,  d,  c and  suppose  the  groups  all 

written  down  ;  then,  fixing  our  attention  upon  any  one  group,  it  is 
evident  that  there  could  be  several  different  arrangements  made 
from  that  group  by  changing  the  order  of  the  letters. 


ii8  Algebra. 

It  is  further  evident  that  if  we  form  all  possible  arrangements  in 
each  group  we  thereby  obtain  the  total  number  of  arrangements 
of  the  71  letters  taken  r  at  a  time. 

The  total  number  of  arrangements  then  equals  the  number  of 
arrangements  in  each  group  multiplied  by  the  number  of  groups. 
Hence,  representing  the  number  of  groups  of  ;/  things  taken  r  at 
a  time  by  GC-)  and  remembering  that  the  number  of  arrange- 
ments in  each  group  equals  the  number  of  arrangements  of  r 
things  taken  all  at  a  time,  that  is  '  r,  and  further  remembering 
that  the  total  number  of  arrangements  equals 

n(n—\)(n—2)  .    .  (7i  —  r^\), 


we  have 


71 

r  G»  =  ,  '- 


hence  GC")  = 


n 


5.  The  form  of  this  result  shows  that  the  number  of  groups  of 
u  things  taken  r  at  a  time  is  the  same  as  the  number  taken  w  — r 
at  a  time.  This  is  also  evident  in  another  way,  for  every  time  we 
select  7  things  from  ii  things  we  leave  out  ;/  —  r  things  ;  hence  there 
must  be  as  many  ways  oi  leaving-  out  w  — r  things  as  of  selectiTtg  r 
things,  but  of  course  there  are  as  many  ways  of  selecting  7i  —  r 
things  as  there  are  of  leaving  out  ii  —  r  things. 

6.  In  all  that  precedes,  it  was  supposed  that  the  given  things 
were  all  different  and  that  in  forming  the  arrangements  or  groups 
none  of  the  given  things  were  repeated.  Now  we  will  consider 
arrangements  and  groups  in  which  the  things  may  be  repeated 
and  those  in  which  the  given  things  are  not  all  alike. 

7.  Problem.  To  find  the  Number  of  Arrangements  op 
71  Things  taken  r  at  a  Time,  Repetitions  being  Allowed. 

Suppose  first  we  wi.sh  the  number  of  arrangements,  including 
repetitions,  of  .the  four  letters  a,  b,  r,  d  taken  one  at  a  time. 
Evidently  there  are  four  arrangements,  viz  :  a,  b,  c,  d. 

Next  suppose  we  wish  the  arrangements,  including  repetitions, 
of  the  four  letters  a,  b,  c,  d  taken  two  at  a  time. 


Arrangements  and  Groups,  119 

The  arrangements  are  the  following  : 
aa  ab  ac  ad 
ba  bb  be  bd 
ca    cb   cc   cd 
da  db  dc  dd 

Thus  we  see  that  there  are  sixteen  arrangements,  that  is,  4^ 
arrangements.  In  exactly  the  same  way  if  we  have  n  letters 
a,  b,  c,  d,  e,f,  .  .  .  ,  the  a  may  be  followed  by  each  of  the  n  let- 
ters, giving  71  arrangements  beginning  with  a  ;  the  b  may  be  fol- 
lowed by  each  of  the  ?i  letters,  giving  n  arrangements  beginning 
with  b,  etc.  So  there  are  evidently  71  arrangements  beginning 
with  eac/i  letter  ;  hence  in  all  there  are  w""  arrangements  of  ti  things 
taken  two  at  a  time,  allowing  repetitions. 

Let  us  now  find  the  number  of  arrangements,  allowing  repeti- 
tions, of  n  things  taken  three  at  a  time;  and  first  to  give  definite- 
ness  to  the  ideas,  consider  the  number  of  arrangements,  allowing 
repetitions,  of  four  letters  a,  b,  c,  d  taken  three  at  a  time.  We 
have  written  out  the  sixteen  arrangements  of  four  letters  taken 
two  at  a  time,  and  now  we  may  suppose  each  of  these  sixteen 
arrangements  to  be  preceded  by  the  letter  a,  then  each  of  these 
sixteen  arrangements  to  be  preceded  by  b,  etc.  We  then  have 
sixteen  arrangements  of  three  letters  each,  beginning  with  each 
letter,  and  as  there  are  four  letters  there  are  in  all  four  times  six- 
teen, or  sixty-four,  arrangements  of  the  letters  a,  b,  c,  d  taken 
three  at  a  time,  repetitions  being  allowed. 

Now,  in  the  same  way,  if  we  have  n  letters  a,  b,  c,  d,  e,f,  .  .  , 
we  may  suppose  each  of  the  n"  arrangements  two  at  a  time  to  be 
preceded  by  a,  then  each  of  these  same  7f  arrangements  to  be  pre- 
ceded b}^  b,  etc. 

Thus  we  get  ir  arrangements  beginning  with  a,  li"  arrange- 
ments beginning  with  b,  if  arrangements  beginning  with  c,  etc. 
Hence  in  all  we  obtain  71  times  ?/^  or  w\  arrangements  of  71  letters 
taken  three  at  a  time,  repetitions  being  allowed. 

hi  ge7ieral,  if  we  know  the  number  of  arrangements  of  71  letters 
taken  5  at  a  time,  repetitions  being  allowed,  we  may  find  the  num- 
ber of  arrangements  of  the  71  letters  taken  ^+1  at  a  time. 

Representing  the  number  of  arrangements,  with  repetitions,  of 


I20  Algebra. 

;/  letters  a,  b,  i\  d,  <?, /,  .  .  .  taken  j-  at  a  time  by  N,,  we  may 
then  write  a  before  each  of  these  N^  arrangements  ^  at  a  time  and 
obtain  N,  arrangements  s-\-\  at  a  time  beginning  with  a. 

We  may  also  write  b  before  each  of  the  same  N.,  arrangements 
and  obtain  N,.  arrangements  ^H-  lat  a  time  beginning  with  b,  and 
so  on  until  each  of  the  n  letters  a,  b,  c,  d,  .  .is  in  turn  placed  be- 
fore each  of  the  N.,  arrangements  ^  at  a  time,  and  we  then  obtain 
?2N,,  arrangements  taken  ^4-i  at  a  time,  repetitions  being  allowed. 

Represent  this  number  by  N,-^  j  and  we  have 

N,.^l=72N,, 

s  being  a  positive  integer  which  may  be  greater  or  less  than  ;z. 

Giving  s  in  turn  all  intermediate  values  from  r—\  down  to  i 
and  remembering  that  the  number  of  arrangements  one  at  a  time 
is  equal  to  ?z,  we  have 

Nl  =  ;^. 
Multiply  these  equals  together  and  cancel  the  common  factors 
and  we  get 

8.  Problem.  To  find  the  Number  of  Groups  of  71  Things 
Taken  r  at  a  Time,  Repetitions  being  Allowed. 

To  prepare  the  way  for  the  general  case  we  begin  with  the 
groups  of  the  four  letters  a,  b,  c,  d  taken  three  at  a  time,  repeti-. 
tions  being  allowed. 

In  this  case  there  are  twenty  groups,  vis: 
aaa  aab  aac  aad  abb 
a  be    abd  aee  aed  add 
bbb    bbe   bbd  bee    bed 
bdd  eee    ced    edd  ddd 
Now  if  in  each  of  these  twenty  groups  we  leave  the  first  letter 
standing  and  advance  the  second  letter  one  step  and  the  third 
letter  two  steps,  we  get  twenty  new  groups  of  the  six  letters  a,  b, 
f,  d,  e,  f,  as  follows: 


Arrangements  and  Groups.  121 

abc  abd  abe  abf  acd 
ace  acf  ade  adf  aef 
bed  bee  bef  bde  bdf 
bef  ede   edf  eef  def 

The  groups  here  written  are  the  groups  of  the  six  letters  a,  b, 
<:,  d^  e,  /,  without  repetitions. 

In  a  similar  manner  we  may  deal  with  the  general  case  of  the 
number  of  groups  of  n  letters  a,  b,  c,  d,  e,f,  .  .  .  taken  r  2X  a. 
time,  repetitions  being  allowed.  L,et  the  number  of  these  groups 
be  denoted  by  n^  and  suppose  them  all  written  down  in  alpha- 
betical order  ;  then  in  eaeh  of  these  groups  keep  the  first  letter  un- 
changed, advance  the  second  letter  one  step,  the  third  letter  two 
steps,  the  fourth  letter  three  steps  and  so  on. 

We  thus  form  n^  new  groups  containing  all  the  letters  the  orig- 
inal ones  contained,  and  r—i  other  letters.  These  new  groups 
are  written  in  alphabetical  order,  because  the  original  ones  were, 
and  by  the  way  in  which  the  letters  have  been  advanced  it  is  evi- 
dent that  no  letter  is  repeated  in  any  one  of  these  new  groups. 

No  two  of  these  new  groups  are  alike,  else  two  of  the  original 
groups  would  have  been  alike. 

Now  since  each  of  these  new  groups  contain  r  of  the  n-\-r — i 
letters  a,  b,  c,  d,  e,  .  .  .  ,  and  since  no  letter  is  repeated  in  any 
group,  and  since  no  two  groups  are  alike,  therefore  these  new 
groups  constitute  some  or  all  of  the  groups  of  the  it  -j-  r—  i  letters 
a,  by  e,  d,  e,  .    .    .  taken  r  at  a  time  without  repetitions. 

Let  the  number  of  groups  without  repetitions  of  n  -f-  r—  i  things 
taken  rat  a  time  be  represented  by  G("+''~^;  then  it  is  evident  that 
N;.  cannot  exceed  G("^''~^).  Now  let  us  conceive  each  of  the 
G("^''~0  groups  written  down  in  alphabetical  order,  and  then 
leave  the  first  letter  in  each  group  unchanged,  change  the  second 
letter  in  each  group  to  the  one  just  before  it  in  the  alphabet,  the 
third  one  in  each  group  to  the  second  one  before  it  in  the  alpha- 
bet and  so  on,  then  these  groups  are  changed  into  new  groups 
wherein  some  of  the  letters  are  repeated,  but  no  letter  is  beyond 
the  wth  letter  of  the  alphabet.  Moreover  no  two  of  these  groups 
are  alike,  ^liCft  no  two  of  those  from  which  they  were  formed 
were  alike,  so  that  these  new  groups  must  be  some  or  all  of  the^^ 
n  letters  a,  b,  c,  d,  e,  .    .    .  taken  r  at  a  time  with  repetitions. 

A— 15 


122  AI.GKBRA. 

These  last  formed  groups  are  G("+''~^)  in  number,  being  formed 
from  that  number  of  groups,  and  as  the  number  of  groups  with 
repetitions  of  n  things  taken  r  at  a  time  has  already  been  repre- 
sented by  N,-,  hence  G(""^"~^)  cannot  exceed  n,.. 

It  was  previously  proved  that  n^  could  not  exceed  G("+'~0» 
hence,  since  neither  can  exceed  the  other,  the  number  must  be  the 
same,  or,  in  other  words,  the  number  of  groups  of  n  things  taken 
r  at  a.  time,  repetitions  being  allowed,  is  equal  to  the  number  of 
groups  of  (?t-\-r—i)  things  taken  r  at  a  time  without  repetitions. 
The  last  number  has  already  been  found.  Hence  the  number  of 
groups  of  n  things  taking  r  at  a  time,  repetitions  being  allowed, 

equals 

(?i-{-r — i)(n-{-r — 2)  .    .    .  n 

k 

which  may  be  written  in  either  of  the  forms 

n(n-\-i)   .    ,    .   (n-hr—i) 


|«  +  r— I 
or  ,"  r^ 


n — I      r 


9.   Problem.     To  find  the  Number  of  Arrangements 
WHERE  THE  Given  Things  are  not  all  Different. 

Illustration. — From  what  has  gone  before  we  know  that  the 
number  of  arrangements  of  the  letters  a,  d,  c,  d  taken  all  at  a  time 
is  twenty-four,  but  if  we  have  the  letters  a,  a,  b,  c  the  number  of 
arrangements  is  only  twelve.     These  twelve  are  the  following  : 
aabc  aacb  abac  abca 
acab  acba  baac  baca 
bcaa  caab  cdba  cbaa 
If  we  have   the  letters  a,  a,  b,  b  there  are  only  six   arrange- 
ments, viz  : 

aabb  abab  abba 
baab  baba  bbaa 
If  we  have  the  letteas  «,  a,  a,  b  there  are  only  four  arrange- 
ments, viz  : 

aaab  aaba  abaa  baaa. 

Thus  we  see  that  with  a  given  number  of  things  the  number  of 
arrangements  depends  upon  how  many  of  each  kind  are  alike. 


Arrangements  and  Groups.  123 

Suppose  now  we  have  in  all  n  letters,  of  which  a  is  repeated  r 
times,  b  is  repeated  s  times,  c  is  repeated  /  times,  and  so  on  so 
that  r-\'S-\-t-\-  .  .  .  —n,  and  we  wish  to  find  the  number  of 
arrangements  taking  all  the  n  letters  at  a  time. 

Fixing  our  attention  upon  any  arrangement  whatever  of  the  n 
letters,  let  all  the  letters  but  the  «'s  remain  unchanged  while  the 
r  a's  change  places  among  themselves.  Because  all  these  <2's  are 
alike  we  get  only  one  arrangement,  but  if  they  had  all  been  dif- 
ferent we  would  have  obtained  \r  arrangements,  and  since  the 
same  thing  is  true  whatever  the  arrangement  upon  which  we 
fixed  our  attention  to  begin  with,  it  follows  that  there  are  |  r 
times  as  many  arrangements  when  all  the  r  letters  are  different  as 
there  are  under  the  present  supposition.  In  the  same  way  there 
are  ■  .y  times  as  many  arrangements  when  the  ^  ^'s  are  all  different 
as  there  are  under  the  present  supposition,  also  there  are  |  /  times 
as  many  arrangements  when  the  t  ds  are  all  different  as  there  are 
under  the  present  supposition,  and  so  on. 

Hence  there  are  1  r  j  ^  \t.  .  .  times  as  many  arrangements 
when  the  n  letters  all  are  different  as  thtre  are  under  the  present 
supposition,  or  the  number  of  arrangements  under  ^^e  present 
supposition  is  equal  to  the  number  of  arrangements  pf  n  things 
taken  all  at  a  time,  when  all  are  different,  divided  hy  \r  Is  \  t  .  .  , 
that  is,  the  number  of  arrangements  under  the  present  supposition 
is  equal  to 


10,  Problem.  To  find  the  Number  of  Ways  in  which 
u  Thing?,  no  Two  Awke,  can  be  Made  up  into  Sets  of 
which  th^  first  set  coji tains  ^  things,  the  second  set  contains  ^  things, 
the  third  contains  /  things,  arid  so  on,  where  of  course 

ir+^  +  /+   .    .    .  =w. 

We  begin  with  a  special  case  and  find  the  nuipber  pf  w^ys  fiy^ 
letters  a,  b,  c,  d,  e,  can  be  wade  up  into  twQ  sets  of  which  the 
first  set  contains  two,  and  the  second  set  three  letters. 

Consider  any  particlar  way  of  dividing  into  sets,  say  the  first 
set  is  ab,  and  the  second  set  is  cde.     Then  keeping  the  sets  undis- 


124  Algebra. 

turbed,  there  could  be  twelve  arrangements  made  from  this  one 

divisions  into  sets.     The  twelve  arrangements  are: 

ab  cde  ba  cde 

ab  ced  ba  ced 

ab  dee  ba  dee 

ab  dee  ba  dee 

ab  eed  ba  eed 

ab  ede  ba  ede 

From  any  other  way  of  dividing  into  sets  there  could  be  twelve 

arrangements  found,  hence  the  whole  number  of  arrangements  of 

five  letters  equals  twelve  times  the  number  of  ways  of  dividing 

into  sets,  or  the  number  of  sets  equals  one- twelfth  the  number 

of  arrangements.  The  number  of  arrangements  in  this  case  is  15, 

hence  the  number  of  ways  of  makimg  up  sets  in  this  case  equals 

tV  |i=io.     ■ 

We  will  now  take  the  general  case  of  n  letters  a,  b,  e,  d,  e,  f,  .    . 
and  take  the  first  r  letters  to  form  the  first  set,  the  following  s  let- 
ters to  form  the  second  set,  the  next  following  t  letters  for  the 
third  set  and  so  on. 

Place  the  letters  of  the  first  set  down  in  a  horizontal  line,  then 
those  of  the  second  set  in  the  same  horizontal  line  followmg  the 
first  set,  and  those  of  the  third  set  in  the  same  horizontal  line  fol- 
lowing those  of  the  second  set,  and  so  on. 

We  thus  have  all  the  n  letters  arranged  in  a  horizontal  line,  and 
it  is  evident  that  we  could  keep  these  sets  undisturbed,  but  still 
make  several  arrangements  of  the  7i  letters  in  a  horizontal  line. 
The  letters  in  the  first  set  can  be  arranged  in  i  r  ways,  those  of 
the  second  set  in  \  s  ways,  those  of  the  third  set  in  |  /  ways,  and 
so  on,  and  as  any  arrangement  in  any  set  may  accompany  atiy 
arrangement  in  a?iy  other  set,  hence  the  whole  number  of  arrange- 
ments while  the  sets  are  undisturbed  is  equal  to  |  r  I  ^  |  /  .    .    . 

Thus  from  one  way  of  making  up  the  sets  there  are  '■  r\  s\  t  .  . 
arrangements,  and  of  course  from  any  other  division  into  sets, 
there  could  be  formed  the  same  number  of  arrangements^  hence 
the  whole  number  of  arrangements  of  n  things,  all  at  a  time, 
equals  \r\s\t  .  .  .  times  the  number  of  ways  of  making  up  the 
sets,  or  the  number  of  ways  of  making  up  the  sets,  equals  the 


Arrangements  and  Groups.  125 

number  of  arrangements  divided  by  |  r  |  ^  |  /  .  .  ,  or  the  number 
of  ways  of  making  up  ?i  things  into  sets,  of  which  the  first  con- 
tains r  things,  the  second  ^  things,  the  third  /  things,  and  so  on, 
equals 

I  ^ 

II.  Given  a  set  of  K  things,  Another  set  of  L  things, 

Another  of  M  things,  and  so  on  ;  to  find  the  Number  of 

Groups  that  can  be  Made  by  taking  r  Things  from  the 

First  set,  5  Things  from  the  Second  set,  /  from  the  Third 

SET,  and  so  on. 

|K 
Of  the  K  things  taken  r  at  a  time  there  are  , — groups, 

I  r  I  K— r  '^ 

and  of  the  L  things  taken  ^  at  a  time  there  are  p  1  y  ~~  groups, 

S         ly  S 

and  of  the  M  things  taken  /  at  a  time  there  are  .  7-1  ?y    ,  groups, 

and  so  on,  and  as  any  one  of  the  groups  from  the  first  set  may  be 
taken  with  any  one  of  the  groups  from  the  second  set,  and  any 
one  from  the  third  set,  and  so  on,  to  form  a  larger  group,  it  follows 
that  the  total  number  of  these  larger  groups  equals  the  product 

|K  IL  IM 


|r|K-r  \s_  |L-^  \^  IM-/ 

12.  There  are  various  relations  connecting  arrangements  with 
arrangements,  groups  with  groups,  arrangements  with  groups, 
etc.  We  will  obtain  a  few  of  these  relations,  and  recommend 
that  the  student  try  to  obtain  others  not. here  given. 

One  relation  was  obtained  in  Art.  2,  where  it  was  shown  that 

A(;:)=«Ac:ii),  (i) 

and  another  in  Art.  3,  where  it  was  shown  that 

Ac;)=;^AC;zl).  (2) 

We  have  already  found  that 

A(")=,i_,  (,) 


126  Algebra. 

and  from  this  it  follows  that 

A("-.)=^j^  (^) 

But  Ui—r-\-i  equals  the  product  of  the  integer  numbers  from 
I  up  to  7i—r-\-i,  and  this  product  of  course  equals  7t—r-\-i 
times  the  product  of  the  integer  numbers  from  i  up  to  «— r,  or 

\7i--^r-\-i  —  (7t—r-\-i)  \n-^r, 

\7l 

hence  A(;.'_i)=-7— t^^t^ (s) 

Comparing  this  with  the  value  of  A("),  equation  (t^),  we  get 
A(';)=r«-r+i;AC_i).  (6) 

If  in  (^)  we  make  r=n  we  get 

AC;)=A(-_,),  (7) 

or  the  number  of  arrangements  of  n  things  taken  all  at  a  time 
equals  the  number  of  arrangements  of  7i  things  all  but  one  at  a 
time. 

We  have  already  found  that 

\7l 


and  from  this  it  follows  that 


G(rO  =  r-^==-— -  (9) 


7i^r—i 


Multiply  both  numerator  and  denominator  of  this  last  fraction 
by  7i(7%^r),  remembering  that  7i  I  ii  —  i  =  I  ;^  and  that  f  ?z — r J  |  n—r^\ 
=??|  i?— r,  ^nd  we  get 


(n—-r)\  71 

n\r  \  n—r  .  ^ 


hence,  from  (%)  and  (c^), 


7i-r-  r 


From  (8  j  it  easily  follows  that 


Cr(n  \  — != .  (\2) 


Arrangements  and  Groups.  127 

Multipl}^  both  numerator  and  denominator  of  this  last  fraction 
by  r  and  remember  that  rVr-^r-^W  and  that  |«— r-f-i«« 
(n—r-\-\)  I  «— r,  we  get 

r\  n 


Comparing  (^13  j  and  (%)  we  easily  get 
From  (%)  it  easily  follows  that 


G(")=^-^'Ga'_,>.  •  ri4; 


From  (\^)  and  (^gj  we  get 

\n—\  \n — I 

r    «— /--f  I        r— I  U^-^r 

(n—r)\n—\         r\n^\  nln^^i  \ii 


n^r  \r   \n — r  r  \  n—r 


v^y 


which  by  Art.  (^)  equals  G("),  hence 

Gc;)=G(r^)+GC;zj).  (1^) 

We  have  obtained  a  few  relations  connecting  arrangements 
with  arrangements  in  equations  (\),  (2),  (6),  (y),  also  a  few 
relations  connecting  groups  with  groups  in  equations  fii),  (14), 
(^i6j.  We  now  obtain  a  few  relations  involving  both  arrange- 
ments and  groups  in  the  same  equation. 

We  have  already  found  in  Art.  4 

AC)=|rG(;),  ^17; 

and  as  I  r=A(r)  we  may  write  (ij)  in  the  form 

Ao-AOG&o.  ris; 

From  (y),  A(;:)=A(;:_i)  and  writing  this  value  in  ('iSj  we  get 

Aao=A(;_i)G(';).  ri9; 

In  (^i8j  substitute  the  valufe  of  G(")  given  in  (16)  and  we  get 

A(';)=A(;)  [G(rM+G(;'z})].  (20J 

But  it  readily  follows  from  (^i8j  that 

A(r^)=A(r)G(rM, 

Substitute  this  value  of  G("~M  ift  (^20^  and  we  get 

A(;')=A(rM+A(;)  G(;'zj).  r2i; 


128  Algebra. 

Since  by  Art.  8,  groups  where  repetitions  are  allowed  can  be 
expressed  in  terms  of  groups  when  repetitions  are  not  allowed,  it 
would  be  an  easy  matter  to  obtain  equations  involving  groups 
with  repetitions. 

13.  Examples  and  Problems. 

1.  How  many  different  groups  of  two  each  can  be  made 
from  the  letters  a,  d,  I,  n,  sf     See  VIII,  Art.  5. 

2.  How  many  arrangements  of  five  each  can  be  made  from 
the  letters  of  the  word  Groups  f 

J.  How  many  different  signals  can  be  made  with  five  flags 
of  different  colors  hoisted  one  above  another  all  at  a  time  ? 

4.  How  many  different  signals  can  be  made  from  seven 
flags  of  different  colors  hoisted  one  above  another,  five  at  a  time  ? 

5.  How  many  different  groups  of  1 3  each  can  be  made  out 
of  52  cards,  no  two  alike? 

6.  How  many  different  signals  can  be  made  from  five  flags 
of  different  colors,  which  can  be  hoisted  any  number  at  a  time 
above  one  another  ? 

7.  How  many  different  signals  can  be  made  from  seven  flags 
of  which  2  are  red,  i  white,  3  blue,  i  yellow  when  all  are  dis- 
played together,  one  above  another,  for  each  signal. 

8.  A  certain  lock  opens  for  some  arrangement  of  the  num- 
bers o,  I,  2,  3,  4,  5,  6,  7,  8,  9,  taken  6  at  a  time,  repetitions 
allowed.  How  many  trials  must  be  made  before  we  would  be 
sure  of  opening  the  lock  ? 

p.  In  how  many  ways  can  a  committee  of  3  appointed  from 
5  Germans,  3  Frenchmen  and  7  Americans,  so  that  each  nation- 
ality is  represented  ? 

10.  How  many  different  arrangements  can  be  made  of  nine 
ball  players,  supposing  only  two  of  them  can  catch  and  one  pitch  ? 

11.  How  many  different  products  of  three  each  can  be  made 
from  the  four  letters  a,  b,  c,  df 

12.  In  how  many  different  ways  can  the  letters  of  the  word 
algebra  be  written,  using  all  the  letters  ? 


Arrangements  and  Groups.  129 

/J.  In  how  many  ways  can  a  child  be  named,  supposing  that 
there  are  400  different  Christian  names,  without  giving  it  more 
than  three  Christian  names  ? 

/^.     In  how  many  ways  can  seven  people  sit  at  a  round  table  ? 
75.     There  are  5  straight  lines  in  a  plane,   no  two  parallel  ; 
how  many  intersections  are  there  ? 

16.  On  a  railway  there  are  20  stations  of  a  certain  class.  Find 
the  number  of  different  kinds  of  tickets  required,  in  order  that 
tickets  may  be  sold  at  each  station  for  each  of  the  others. 

77.  Find  the  number  of  signals  that  can  be  made  with  four 
lights  of  different  colors,  which  can  be  displayed  any  number  at 
a  time,  arranged  either  above  one  another,  side  by  side,  or 
diagonally. 

18.  From  a  company  of  90  men,  20  are  detached  for  mount- 
ing guard  each  day ;  how  long  will  it  be  before  the  same  20  men 
are  on  guard  together,  supposing  the  men  to  be  changed  as  much 
as  possible  ?  How  often  will  each  man  have  been  on  guard  during 
this  time  ? 

ig.  A  lock  contains  5  levers,  each  capable  of  being  placed  in 
10  distinct  positions.  At  a  certain  arrangement  of  the  levers  the 
lock  is  open.  How  many  locks  of  this  kind  can  be  made  so  that 
no  tw^o  shall  have  the  same  key  ? 

20.  There  dre  n  points  in  a  plane  no  three  of  which  are  in 
the  same  straight  line.  Find  the  number  of  straight  lines  which 
result  from  joining  them. 

21.  There  are  n  points  in  a  plane,  no  three  of  which  are  in 
the  same  straight  line  except  r,  which  are  all  in  the  same  straight 
line  ;  find  the  number  of  straight  lines  which  result  from  joining 
them. 

22.  There  are  n  points  in  space,  no  four  of  which  are  in  the 
same  plane  with  the  exception  of  r  which  are  all  in  the  same 
plane.  How  many  planes  are  there,  each  containing  three  of  the 
points  ? 

A— 16 


CHAPTER  X. 

BINOMIAI.   THEOREM. 

1.  The  Binomial  Theorem  enables  us  to  find  any  power  of  a 
binomial  without  the  labor  of  obtaining  the  previous  powers.  In 
order  to  observe  the  law  of  formation  of  a  power  of  a  binomial  we 
first  observe  the  law  of  formation  of  the  product  of  several  binomial 
factors  of  the  form  x+a,  x-\-d,  x-\-c,  etc.,  and  we  will  afterwards 
be  able  to  arrive  at  the  power  of  a  binomial  by  the  supposition 
that  a=b=c,  etc. 

2.  IvAw  OF  THE  Product  of  Factors  of  the  form  x-\-a, 
x+d,  x-\-c,  Etc. 

By  actual  multiplication  it  is  seen  that 
(x-\-a)(x-\-b)=x'-\-(a-\-b)x-\-ab, 

(x-\-a)(x-\-b)(x-\-c)=x^-\-(a-\-b-\-c)x^-\-(ab-\-ac-\-bc)x-{-abc, 
(x+a)(x+b)(x+c)fx-\-d)=x'-{-(a-\-b-j-c-j-d)x'  + 

(ab  +  ac-\-ad-\-  bc-\-  bd-\-  cd)x-  +  (abc-\-abd-\-  acd-\-  bcd)x  -\-abcd. 
By  a  careful  inspection  of  these  products  we  will  discover  the 
presence  of  two  uniform  laws  —  a  law  for  the  exponents  and  a  law 
for  the  coefficients. 

The  law  of  the  exponents  is  readily  seen  to  be  as  follows  : 
The  expoyient  of  x  in  the  first  term  of  the  product  is  equal  to  the 
number  of  binomial  factors,  and  in  the  remining  terms  it  continually 
decreases  by  07ie  until  it  is  zero. 

The  law  of  the  coefficients  may  be  stated  thus  : 
7^he  coefficient  of  the  first  term  is  unity;  the  coefiicicnt  of  the  second 
term  is  the  sum  of  the  second  terms  of  the  binomial  factors;  the  coef- 
ficient of  the  third  term  is  the  su?n  of  all  their  different  products 
take7i  two  at  a  time;  the  coefficient  of  the  fourth  term  is  the  sum  of  all 
their  different  products  taken  three  at  a  time,  and  so  on.  The  last 
term  is  the  product  of  all  the  second  terms  of  the  binomial  factors. 

3.  Proof  that  the  Laws  are  General.  We  will  now  show 
that  if  the  laws  observed  above  hold  in  the  product  of  a  given 
number  of  binomial  factors,  they  will  hold  in  the  product  of  any 
number  of  binomial  factors  whatever. 


BiNOMiAi^  Theorem.  131 

For,  assume  that  we  have  tested  the  above  laws  in  the  case  of 
the  product  of  a  certain  number  of  factors,  suppOvSe  n,  and  have 
found  them  to  hold  true. 

To  facilitate  the  discussion  we  will  reprCvSent  the  n  second  terms 
of  the   binomial    factors   by  a^,  a,^,  a^,  a^,  .    .    .  «„*  instead  of 
a,  d,  c,  d,  etc.,  and   accordingly  the  product  of  the  /i  binomials 
(x~\-aJ(x-\-a.J(x-\-a^)(x-\-a4)  .    .    .  (x-\-a„_-^)(x-\-aJ 
=x"-\-(a^-\-ao-{-a..-\-a^  .    .  -}-a„)x"~'^ 

-\-(a-^a.^-^a^a^-\-a^a^-\-  .    .  -\-a„_^a„)x''~'^ 

In  order  to  abreviate  this  expression  it  is  convenient  to  let 
Pi  equal  the  Jirs^  parenthesis,  or  the  sum  of  all   the   different 

second  terms  of  the  binomial  factors. 
P2  equal  the  second  parenthesis,  or  the  sum  of  all  the  different 

products  of  the  second  terms  of  the  binomial  factors  taken  two 

at  a  time. 
P3  equal  the  third  parenthesis,   or  the  sum  of  all  the    different 

products  of  the  second  terms  of  the  binomial  factors  taken  three 

at  a  time  ;  and  so  on. 
P„  equal  the  ??  th  parenthesis,   or  the  product  of  all  the  second 

terms  of  the  binomial  factors.  • 

With  these  abbreviations  the  second  member  of  the  above 
equation  reads 

jr"  +  PiX"-i+P2-^-"~'  +  P3^"-^-f  .    .    .  +P„. 

Multiplying  this  expression,  which  represents  the  product  of  n 
binomial  factors,  by  a  new  binomial,  x-{-a„^^,  we  derive  the  fol- 
lowing result  for  the  product  of  ;^-f  i  binomial  factors  : 
x''-^^-^(F,-ha„,Jx"-^(F,-\-a„^,FJx"-'^ 

+  (F^-{-a„,,FJx''-''-\-  .    .    .  H-a„,-iP.,. 

It  is  seen  from  this  result  that  the  law  of  exponents  still  holds. 
For  there  are  n-\-i  binomials  and  the  exponent  of  x  begins,  in 
the  first  term,  with  n-\-i  and  decreases  continually  by  one  through 
the  remaining  terms  until  the  value  zero  is  reached. 

♦This  notation  presents  many  mechanical  advantages.  It  must  not  bo  supposed, 
however,  that  there  is  any  relation  subsisting  between  n^  and  a.2  or  any  other  two  of  the 
symbols ;  they  are  as  independent  as  distinct  letteis. 


132  Algebra. 

The  law  of  coefficients  holds  good  also.     For  : 

The  coefficient  of  the  first  term  is  unity. 

The  coefficient  of  the  second  term  is  F^-\-a„.^.  Now  P^ 
stands  for  the  sum  of  the  n  terms  a^-^a^-i-a^-l-.  .  .  a„.  Hence 
Pj-f  <2„^  1,  or  the  coefficient  of  the  second  term,  is  the  sum  of  all 
the  different  second  terms  of  the  binomial  factors. 

The  coefficient  of  the  third  term  is  P2+«;h  i^^i- 

Now  P2  represents  the  sum  of  all  the  different  products  of  the 
?t  letters  a^,  a2,  a^,  a^^,  .    .    .  a„  taken  two  at  a  time. 

(i).  That  is,  P2  represents  the  sum  of  all  the  different  products 
of  the  n-\-i  letters  a^,  a^,  a.,^,  .  .  .  a,,^^  taken  two  at  a  time 
which  do  not  contain  «„+i . 

Again, 

(2).  That  is,  ^„+iP2  equals  the  sum  of  all  the  different  prod- 
ucts of  the  7^-f-I  letters  a^,  a.^_^,  a^^,,  a^,  .  .  .  a,^^^,  taken  two  at  a 
time,  which  contain  a,,;^. 

Therefore,  putting  (i)  and  (2)  together,  P2H-^„.  iPi  equals 
the  sum  of  all  the  different  products  of  the  ?i  -f  i  letters 
«i,  <2  2,  ^3,  «4,  .  .  .  «„4-i,  taken  two  at  a  time,  both  those  which 
do  and  those  which  do  not  contain  ^„+i. 

Th^  coefficient  of  the  fourth  term  is  Pg+a,,^  1P2. 

Now  P3  equals  the  sum  of  all  the  different  products  of  the  ?i 
letters  a^,  a^,  a^,  a^,  .    .    .  a,„  taken  three  at  a  time. 

(^3J.  That  is,  P3  equals  the  sum  of  all  the  different  products  of 
the  ?i-\-i  letters  «i,  a^,  a^^,  a^,  .  .  .  a„.^.^,  taken  three  at  a  time, 
which  do  not  contain  a,,.^^. 

Again, 

«„+lP2=«i«2^«-M+«1^3^«+l+^1^4^«fl+    •      •    +'^;/-l^"^//l  1- 

(\).  That  is,  «„4 1P2  equals  the  sum  of  all  the  different  products 
of  the  n-\-\  letters  a^,  a^,  a^,  a^,  .  .  .  a,,_,_^ ,  taken  three  at  a  time, 
which  contai7i  a,,^.^. 

Therefore,  putting  (t,)  and  (4.)  together,  Pi^+a„_^^F2  equals 
the  sum  of  all  the  different  products  of  the  ?i+i  letters  a-^^,  a^, 
a^,  <24,  .    .    .  «„-;  I,  taken  three  at  a  time. 

In  like  manner  we  may  treat  the  coefficient  of  the  fifth  term, 
and  so  on.     The  last  term  is  the  product  of  all  the  n-\-i  letters 

«!,   ^2)   ^3»   «4.    •      •      •   <^«+l- 


Binomial  Theorem.  133 

Therefore,  we  have  proved  that  if  the  laws  of  exponents  and 
coefficients  hold  in  the  product  of  n  factors,  they  will  hold  also  in 
the  product  of  ^2+  i  factors. 

But  they  have  been  proved  by  actual  multiplication  to  hold 
when  four  factors  are  multiplied  together,  therefore  they  hold 
when  five  factors  are  multiplied  together,  and  if  they  hold  when 
five  factors  are  multiplied  together  they  must  hold  when  six  are 
multiplied  together,  ana  so  on  indefinitely.  Hence  the  laws  hold 
universally, 

4.  Deduction  of  the  Binomial  Formula. 
We  have  now  proved  that  the  equation 

(x-\-a^)(x-\-a^J(x-\-ar,)  .    ,    .  (x-^a„._^)(x-\-aJ 
=  x"  +  (a,+a.,-\-a.,+  .    .    .  +ajx"-^ 

'\-(a^a2-{-ci-^a.;^-\-a-^a^-\-  .    .    .  -\-a„_-^a,Jx"~'^ 

is  true  for  all  positive  values  of  n. 

Since  a^,  a2,  a.^,  a^,  .  .  .  a„  are  any  numbers  whatever,  we 
may  assume  that  they  are  all  alike  and  we  may  suppose  each 
equal  to  the  quantity  a.  Then  each  of  the  factors  in  the  left- 
hand  side  of  the  above  equation  will  become  equal  to  x-\-a,  and 
consequently  the  left-hand  member  will  become 

On  the  right-hand  side  of  the  equation  the  term  x"  remains  un- 
changed. I^n  the  second  term  the  parenthesis  becomes  the  sum 
of  n  a's  ;  that  is,  it  is  equal  to  ??a,  so  that  the  second  term  itself 
becomes 

nax"~^ . 

In  the  third  term  the  parenthesis  reduces  to  the  sum  of  as  manj^ 

<2-'s  as  there  are  groups  of  71  things  taken  two  at  a  time  :  that  is, 

7i( n —  I  ) 

the  parenthesis  becomes <J^  so  that  the  third  term  itself 

1.2 

becomes 

1.2  ♦ 

In  the  fourth  term  the  parenthesis  reduces  to  the  sum  of  as 
many  «^'s  as  there  are  groups  of  7i  things  taken  three  at  a  time  ; 


134  Algebra. 

7l( 71  —  \  )( 71  —  2  ) 

that  is,    the  parenthesis  becomes   «',   so  that   the 

fourth  term  itself  becomes 

1.2.3 
and  so  on  for  the  other  tenns. 

The  last  term  reduces  to  the  product  of ;/  a's  ;  that  is,  to 

a" . 
Therefore,  on  the  supposition  that  a=a=a^—  .    .    .  =a„,  the 
equation  above  written  becomes 

(x — a)"= 

1.2  1.2.3 

which  is  the  Bino77iial  Formula. 

The  expression  on  the  right-hand  side  of  the  equation  is  called 
the  Expa7isio7i  or  the  Develop7ne7it  of  the  power  of  the  binomial. 

5.  Example.   Expand  (y-\-2)^. 

Substitute  r  for  x,  2  for  a,  and  5  for  71,  in  the  binomial  formula 
and  we  obtain 

I  .  2  1-2.3  '  ^-2.3.4 

or  simplifying, 

(y-\-  2>^=r5-f  ioy-f4oy  4- 8oj'^-f  80^+32. 

6.  Binomial  Theorem.  The  binomial  formula  may  be  stated 
in  the  form  of  a  theorem  as  follow^s  : 

hi  a7iy  poiver  of  a  bi7i077iial  x-\-a,  the  exp07ient  of  x  begi7is  171  the 

first  term  with  the  expohcTit  of  the  poiver,  a7id  in  the  followi7ig  ter77is 

co7iti7iually  decreases  by  one.      The  exp07ie7it  of  a  com77iences  u'ith 

07ie  in  the  seco7id  ter77i  of  the  power,  a7id  co7iti7iiially  i7icreases  by  one. 

The  coefficie7it  of  the  first  te7'7n  is  07ie,  that  of  the  seco7id  is  the  ex- 
p07ie7it  of  the  power  ;  and  if  the  coefficie7it  of  any  ter7n  be  77iultiplied 
by  the  expo7ie7it  of  x  i7i  that  ter77t  and  divided  by  the  exp07ient  of  a 
i7icreased  by  07ic,  it  will  give  the  coefficie7it  of  the  siicceedi7ig  term. 

7.  Historic aIj  Note.  The  first  rule  for  obtaining  the  powers  of  a  bi- 
nomial seems  to  have  b(?en  given  by  Vieta  (1540-l{i03).  He  observed  as  a 
necessary  result  of  the  process  of  multiplication  that  the  successive  coefficients 
of  any  power  of  a  binomial  are  :  first,  unity  ;  second,  the  sum  of  the  first  and 


BiNOMiAi.  Theorem.  135 

secon(i  coefficients  in  the  preceding  power ,  third,  the  sum  of  the  second  and 
third  coefficients  in  the  preceding  power,  and  so  on.  Vieta  noticed  also  tlie 
uniformity  in  the  product  of  binomial  factors  of  the  form  x-\-(i,  x-\-h,  x-\-c,  etc. 
But  Harriot  (1560-1621)  independently  and  more  fully  treated  of  these  prod- 
ucts iu  showing  the  nature  of  the  composition  of  a  rational  integral  equation. 
See  VI,  Art.  1.  In  this  connection  it  is  interesting  to  note  that  Harriot  was 
the  first  mathematician  to  transpose  all  the  terms  of  an  equation  to  the  left 
member. 

The  binomial  formula  as  now  used  ;  that  is,  the  expansion  of  the  n  th  power 
of  a  binomial,  expressed  with  factorial  coefficients,  was  the  discovery  of  Sir 
Isaac  Newton  (1642-1727)  and  for  that  reason  it  is  commonly  called  Sir  Isaac 
Newton  s  Binomial  Theorem. 

8.  Number  of  Terms  in  the  Expansion.     The  exponents 

of  «  through  the  binomial  formula  constitute  the  following  scale  : 
o,  I,  2,  3,  4,  .    .    .  ?i. 
The   number  of  terms  in   this  scale  is   7i-\-i.     Therefore    the 
number  of  terms  in  the  expansion  of  (x-\-a)"  is  ?^+  i. 

.9.  Value  of  the  mi  Term.  The  value  of  the  rth  term  in 
the  expansion  of  (x-\-a)"  can  be  easil}^  found 

By  the  law  of  exponents,  the  exponent  of  j;  in  the  /irs^  term  is 
;^  ;  in  the  second,  n — i  ;  in  the  third,  n  —  2,  and  so  on;  conse- 
quently in  the  rth  term  it  is  n  —  (r—\),  or  ??  — r-f  i.  Also  by  the 
law  of  the  exponents,  the  exponent  of  a  in  the  second  term  is  i  ; 
in  the  third  term,  2,  and  so  on  ;  conseqtiently  in  the  rth  term  it 
is    r— I.     So,    without   the    coefficient,    the    rth    term    must   be 

By  inspection  of  the  coefficients  in  the  expansion  in  Art.  4, 
it  is  seen  that  the  numerator  of  the  coefficient  of  any  term  is  the 
product  of  the  natural  numbers  from  ;/  to  a  number  one  greater 
than  the  exponent  of  a.  Since  the  exponent  of  .r  in  the  rth  term 
has  been  found  to  be  ;?— r-fi,  this  numerator  of  the  coefficient 
vciVL^\,h^  7i(n—\)(n—2)  .    .    .  (7i  —  r-\-2). 

An  inspection  of  the  binomial  fonnula  will  also  show  that  the 
denominator  of  any  coefficient  is  the  product  of  the  natural  num- 
bers from  unity  to  a  number  equal  to  the  exponent  of  a.  Whence 
the  denominator  of  the  coefficient  of  the  rth  term  must  be 
I,  2,  3,  .  .  .  (r—\).  Therefore  the  complete  ?'th  tenn  is 
71(11— \)  (11  —  2)  .  .  .  (n  —  r-^2) ^,._^  ^  „._^., , 
1.2.3.4  .    .    •  i^'—"^') 


136  Algebra. 

Multiplying   numerator   and   denominator  of  the  coefficient  by 
I  n — r-f  I ,  this  becomes 

r—  I      \n  —  r+  i 

10.  Theorem,  hi  the  expansion  of  (x^ a)"  the  coeffide^it  of 
the  rth  term  from  the  beginyiing  equals  the  coefficient  of  the  rth  teim 
from  the  end. 

Since  there  are  ?z+i  terms  all  together  (Art.  8),  the  /th  term 
from  the  end  has  7i-\-\—t,  or  n—t-\-\,  terms  before  it.  Hence 
the  /th  term  from  the  end  is  the  same  as  the  ;^  — /4-2th  term  from 
the  beginning.  From  the  preceding  article  the  ;z— /-|-2th  term 
equals 


\n—t-\-\     \t—\ 

But  from  the  preceding  article  the  /th  term  from  the  beginning 
equals 


|/— I      \^n  —  t-^\ 
It  is  plainly  seen  that  the  coefficients  are  identical. 

11.  Expansion  of  (x—a.)" 

If  we  substitute  —a  for  a  in  the  binomial  formula^we  will 
obtain  the  following  result  for  the  expansion  oi  x—a  : 

(x—a)"=x"~nax"-'-\--  ^a\x"'' *  ^'  \a^x"-^4-  .   . 

1.2  1.2.3 

12.  Theorem.  /?i  the  binomial  formula  the  s?im  of  the  coeffi- 
cients of  the  even  terms  equals  the  sum  of  the  coefficients  of  the  oda 
terms. 

In  the  expansion  of  (x—a)"^\x\.  x=i  and  a=\.  We  then 
obtain 

,  n(n—i)     7i(n—i)(n—2)  , 

0=  I  —  ?z  -f  -^ ^ <^ ^  4-  etc. , 

1.2  1.2.3 

which  shows,  since  the  negative  on  the  right  side  of  this  equation 

must  equal  the  positive,  that  the  sum  of  the  coefficients  of  the 

first,  third,  fifth,  .    .    .  terms  equals  the  sum  of  the  coefficients  of 

the  second,  fourth,  sixth,  .    .    .  terms. 


Binomial  Theorem.  137 

13.  Theorem.    The  sum  of  all  the  eoejjicients  in  the  expansion 
0/  (x-\-a)"  equals  2". 

In  the  expansion  of  (x-\-a)"  ^nt  x—i   and  a=\.     We  then 

have 

n(n—i)     n(n—i)(?i—2) 

2=1+;/+     -  +-  +etc, 

2  1-2.3 

14.  Examples. 

/.  Expand  (x-\-a)^. 

2.  Expand  (b—e)^. 

J.  Expand  6'+3/- 

^.  Expand  (b'—e")^. 

5.  Expand  (x-\-af. 

6.  Expand  (x-\-2c)^. 

7.  Expand  (2>b+i)\ 

8.  Expand  (x'-\-a')\ 
g.  Expand  (2ax—x')\ 

10.  Expand  Wab—^abf- 

11.  Expand   y-i--\  • 

12.  Expand  (5—ix)'\ 

13.  Find  the  5th  term  of  (xy-\-x"-y\ 

14.  Find  the  9th  term  of  [^x'^+^x 

I           I  ]" 
75.     Find  the  ?iih.  term  of  \n"-\ \   . 

{.  'M 

16.     Expand  (x''-\-2ax-\-a'')^. 

ly.     Expand  (V<^^— 2 J^)'^ 

18.     Find  the  1000  term  in  (x-\-ay"^\ 

15.  Expansion  OF  A  Polynomial.  The  power  of  a  polynomial 
can  be  obtained  in  the  following  manner.  Suppose  it  is  required 
to  expand  (a-\-b-^cy.     We  can  proceed  thus  : 

(a^b^ey^{a^(b-^c)J 
=  a'  +  2>a^(b  +  e)  +  2>a(b+e)^-\-(b-\-e)\ 
which,  when  the  powers  of  ^  +  <r  are  developed,  becomes 

a3-f-  3rt!-<^-f  3^V+  2>ab'-\-6abe^  3^^'  +  <^'+  3^V+  3^r" +r^ 
Notice  that  the  result  is  a  homogeneous  symmetrical  fu7iction  of 
a,  b,  and  c. 

—17 


138 


Algebra. 


16. 


ExAMPIvKv^i. 

Expand  (a-^d—c)\ 
Expand  (ad-{-dc+ac)^. 

Expand 


X — a 

Expand  (a-\-d-^t^-\-d)\ 
Expand  (i-^x+x")'. 


CHAPTER  XI. 

THEORY  OF   UMITS. 

1.  Definition.  When  a  quantity  preserves  its  value  un- 
changed in  the  same  discussion  it  is  called  a  Constant,  but  when 
under  the  conditions  of  the  problem  a  quantity  may  assume  an 
indefinite  number  of  values  it  is  called  a  Variable. 

Constants  are  usually  represented  by  the  first  or  intermediate 
letters  of  the  alphabet  and  variables  by  the  last  letters. 

The  notation  by  which  we  distinguish  between  constants  and 
variables  is  the  same  as  that  by  which  we  distinguish  between 
known  and  unknown  quantities,  but  it  must  not  be  thought  that 
any  analogy  is  intended  to  be  pointed  out  by  this  fact.  When  we 
are  discussing  a  problem  in  which  both  constants  and  variables 
appear  we  usually  do  not  care  whether  the  constants  are  known 
or  unknown. 

2.  Definition.  When  a  variable  in  passing  from  one  value  to 
another  passes  through  all  intermediate  values  it  is  called  a 
contmuous  variable;  when  it  doe's  not  pass  through  all  intermed- 
iate values  it  is  called  a  discontinuous  variable. 

3.  Definition.  When  a  variable  so  changes  in  value  as  to 
approach  nearer  and  nearer  some  constant  quantity  which  it  can 
never  equal,  3^et  from  which  it  may  be  made  to  differ  by  an 
amount  as  small  as  we  please,  this  constant  is  called  the  Limit  of 
the  variable. 

4.  Illustrations.    If  a  point  move  along  a  line  AB,  starting 

A  B 

■I       •  II 

at  A  and  moving  in  such  a  wav  that  the  first  second  the  point 

moves  one-half  the  distance  from  A  to  B,  the  second  second  one- 
half  the  remaining  distance,  the  third  second  one-half  the  distance 
which  still  remains,  and  so  on  ;  then  the  distance  from  A  to  the 
moving  point  is  a  variable  whose  limit  is  the  distance  AB.  For, 
no  matter  how  long  the  point  has  been  moving,  there  is  still  some 


140  Algebra. 

distance  remaining  between  it  and  the  point  B,  so  that  the  dis- 
tance from  A  to  the  moving  point  can  never  equal  AB,  but  as  the 
moving  point  can  be  brought  as  near  as  we  please  to  B,  its  dis- 
tance from  A  can  be  made  to  differ  from  the  distance  AB  by  an 
amount  as  small  as  we  please. 

Thus  we  see  that  the  distance  from  A  to  the  moving  point  ful- 
fills all  the  requirements  of  the  definition  of  a  variable,  and  the 
distance  AB  all  the  requirements  of  the  definition  of  a  limit. 

The  student  must  note  that  it  is  not  the  point  B  that  is  the 
limit  of  the  moving  point,  although  the  moving  point  approaches 
the  point  B,  but  it  is  the  distance  AB  that  is  the  limit  of  the 
distance  from  A  to  the  moving  point. 

If  we  call  the  distance  the  point  moves  the  first  second  i  (then 
of  course  the  whole  distance  AB  would  be  2),  the  distance  trav- 
ersed the  second  second  would  be  \,  that  traversed  the  third  .second 
would  be  \,  and  so  on,  and  the  entire  distance  from  A  to  the 
moving  point  at  the  end  of  n  seconds  would  be  the  sum  of  n 

terms  of  the  series 

T    1    1    1     1 

^»    2»    4'    8"'    16"'     •      •      • 

Now  it  is  sure  that  the  more  terms  of  this  series  that  are  taken 
the  less  does  the  sum  differ  from  2  ;  but  the  sum  can  never  equal 
2.     Hence  we  say  that  the  limit  oi  the  sum  of  the  series 

I+-2"+4+"8+T(5"    •      •      • 

as  the  number  of  terms  is  indefinitely  increased  is  2. 

Again  consider  any  regular  polygon  inscribed  in  a  circle,  and 
then  join  the  vertices  with  the  middle  points  of  the  arcs  subtend- 
ing the  sides,  thus  forming  another  regular  inscribed  polygon  of 
double  the  number  of  sides.  From  this  polj^gon  form  another  of 
double  its  number  of  sides  and  so  on.  Now  the  polygon  is  always 
ivithi7i  the  circle,  and  hence  the  area  of  the  polygon  can  never 
equal  the  area  of  the  circle,  but  as  the  process  of  doubling  the 
number  of  sides  is  continued,  the  less  does  the  area  of  the  poly- 
gon differ  from  the  area  of  the  circle.  Hence,  we  say  that  the 
limit  of  the  area  of  the  polygon  is  the  area  of  the  circle. 

Again  as  a  straight  line  is  the  shortest  distance  between  two 
points,  any  side  of  the  inscribed  polygon  is  less  than  the  sub- 
tended arc,  hence  the  sum  of  all  the  sides  or  the  perimeter  of  the 
polygon  is  less  than  the  sum  of  all  the  subtended  arcs  or  the  cir- 


Theory  of  Limits.  141 

cumference  of  the  circle,  or  in  other  words  the  perimeter  of  the 
polygon  can  never  equal  the  circumference  of  the  circle,  but  as 
the  process  of  doubling  the  number  of  sides  is  continued,  the 
perimeter  of  the  polygon  differs  less  and  less  from  the  circumfer- 
ence of  the  circle,  hence  the  circumference  of  the  circle  is  the 
limit  of  the  perimeter  of  the  inscribed  polygon. 

5.  The  student  should  not  infer  from  what  has  been  said  that 
all  variables  have  limits!  In  fact,  the  truth  is  quite  the  contrary, 
for  most  variables  do  not  have  limits.  Thus,  in  the  illustration  of 
the  moving  point  given  above,  the  variable  does  not  have  a  limit 
if  we  suppose  the  point  to  move  at  a  uniform  rate.  For,  if  the 
velocity  is  uniform,  it  is  a  mere  question  of  time  until  the  moving 
point  passes  B,  or,  in  fact,  any  other  point  to  the  right  of  B,  how- 
ever remote.  Much  more  would  this  be  true  if  the  point  moved 
with  increasing  instead  of  uniform  velocity. 

Again,  consider  the  fraction 

X 

If  X  be  supposed  to  change  in  value,  the  value  of  the  fraction 
changes  and  is  itself  a  variable.  Now  suppose  x  to  decrease  in 
value.  It  is  plain  \)ci.2i\.  the  value  of  the  fraction  increases  without 
limit  as  x  decreases.  In  other  words,  the  value  of  the  fraction  can 
be  made  as  large  as  we  please  by  taking  x  small  enough.  Hence, 
as  X  decreases,  the  value  of  the  fractioyi  has  710  limit. 

6.  It  follows  immediately  from  the  definition  of  a  variable, 
that  the  difference  betiveen  a  variable  and  its  limit  is  a  variable 
ivhose  limit  is  zero. 

For  if  X  be  a  variable  whose  limit  is  a,  then  x  may  be  made  to 
differ  from  a  by  as  small  a  quantity  as  we  please,  hence  a—x  may 
be  made  as  small  as  we  please;  yet  as  x  can  never  equal  a,  a—x 
can  never  equal  zero,  hence  a—x  is  a  variable,  whose  limit  is  zero. 

7.  Theorem.  //*  two  variables  are  always  equal  a?id  each 
approaches  a  limit,  the  limits  must  be  equal. 

Let  X  and  v  be  the  variables,  and  let  limit  x=a  and  limit  r=^. 


42 


Algebra. 


We  are  to  prove  that  a^b.     If  a  and  b  are  not  equal,  suppose  a 

greater  than  b  and  let 

a-b=d 

I^et  a—x=u  and  b—y=^i\ 

then  a=x-{-u  and  b=^y-\-i\ 

and  a  —  b=d  becomes  by  substitution, 

(x  +  u)-(y+v)==d 
or  (^.r  —y)-\-(  u —v)  =  d 

Since  lim  x^=a,  lim  u=o  and  as  lim  y=b,  lim  t'=^,  or  ?/;  and  -v 
are  each  variables  which  can  be  made  as  small  as  we  please,  and 
hence  the  difference  u  —  v  can  be  made  as  small  as  we  please,  and 
so  can  be  made  so  small  as  not  to  cancel  d,  hence  x—y  would 
equal  something,  or  x  andjF  would  differ,  which  is  contrary  to  the 
hypothesis;  hence  a  cannot  be  greater  than  b,  and  in  the  same 
way  it  may  be  shown  that  b  cannot  be  greater  than  a. 

Therefore  a^=b. 

8.  Theorem.  The  limit  of  the  algebraic  sum  of  several  variables 
equals  the  algebraic  sum  of  their  separate  limits. 

Let  the  variables  be  x,y,  z,  etc.,  and  let  lim  .v=<7,  lim  r=<^, 
lim  .5-=^,  etc.,  we  are  to  prove 

lim  ^r+j/+2-t-  .    .    .  ;  =  C^  +  ^+r+  .    .    .  j 

I^et  a — x=^u  .'.  x=a  —  u, 

etc. ,  etc. ; 

thenjt-+r  +  -+  .    .    .  =(a-\-b-\-c+  .    .  )-(u  +  v-h7C'+  .    .  ;. 

Suppose  i/  to  be  numerically  the  greatest  of  the  quantities 
u,  V,  zv,  .    .    .  and  suppose  that  there  are  n  of  these  quantities. 

Now,  since  x  may  be  taken  so  near  a  as  to  differ  from  it  by  an 
amount  as  small  as  we  please,  we  may  take  x  so  that 

k 
n 
or  nu<^k, 

however  small  k  may  be. 

Then    u-{-v-\-w-\-  .    .    .  <;^^/   (since    u    is    the    largest   of  the 

quantities  u,  v,  u\  .    .    .  ).     Hence  u -\- v -\- zu -\-  .    .   .</(',  however 

small  k  ma}'  be  ;  that  is,  x-\-y-\-z-\-  .    .    .  may  be  made  to  differ 

fram  a-\-b-\-c  .    .    .by  an  amount  as  small  as  we  please.     Hence 

limit  (x-\-y-\-.c-{-  .    .    .  )  =  a-\-b-\-c-{-  .    .    . 


Theory  of  Limits.  143 

.9  Theorem.  Tlie  limit  of  a  constant  niultiple  of  a  variable 
equals  that  constant  multiplied  by  the  limit  of  the  variable. 

Let  X  be  a  variable  and  a  its  limit.  We  are  to  prove 
lim  nx=7ia. 

Let  a—x=-u.     Then  x  may  be  taken  so  near  to  a  as  to  make 

k 

u<i  -, 
n 

or  nu<ik\ 

however  small  k  may  be. 

a — x=u  .'.  na  —  nx=^nu, 

hence  nx  may  be  made  to  differ  from  na  by  an  amount  as  small 

as  we  please.     Yet  nx  can  never  equal  7ia\  else  x  could  equal  a. 

Hence  lim  nx=na. 

10.  Theorem.  71ie  limit  of  the  pfvduet  of  tzvo  variables  equals 
the  product  of  their  limits. 

With  the  same  notation  as  in  Art    7  we  are  to  prove  that 

lim  xy=ab. 
xy=(a — u)(b — v)=ab — av — bu-\-uv 
=^ab—(av-\-bu — 2iv). 
Since  lim  z'=o,  lim  az'=o, 

and  as  lim  ?/=o,  lim  bu=^o, 

and  since  u  and  v  are  each  as  small  as  we  please  and  the  product 
smaller  than  either,  lim  uv=o.  And  since  the  limit  of  each  term 
oi  av-\-bu  —  uv  is  zero,  the  limit  of  the  algebraic  sum  of  all  three 
terms  is  zero.  Hence  xy  may  be  made  to  differ  from  ab  by  an 
amount  as  small  as  we  please  ;  hence  lim  xy=ab. 

11.  Theorem.  77ie  limit  of  the  product  of  any  number  of  var- 
iables is  equal  to  the  product  of  their  limits. 

With  the  same  notation  as  before  we  are  to  prove 

lim  (xyz  .    .    .  )  =  abc  .    .    . 
We  have  already  proved  that 

lim  xy=ab, 
and  we  may  consider  xy  a  single  variable  and  ab  its  limit ;  then 
by  the  last  article 

lim  (xy.£j===.ab.c, 
or  lim  xy2=adc, 


144 


Algebra. 


and  now  xvz  may  be  considered  a  single  variable  and  abc  its 
limit,  and  a  repetition  of  the  application  of  the  theorem  of  the  last 
article  would  show  that  the  limit  of  the  product  oi  four  variables 
equals  the  product  of  their  limits,  and  evidently  this  reasoning 
could  be  carried  as  far  as  we  wish. 

12.  ThkOKEM.    The  limit  of  the  quotient  of  two  variables  equals 

the  quotient  of  theif  limits. 

With  the  same  notation  as  before,  we  are  to  prove  that 

,.      X     a 
lim  "=7". 

^       X  . 

Let  — =<7,  then  x=^qy. 

y 

.'.  lim  -r=lim  (qy)  =  lini  qAimy; 
..  lim  X     a 

lim  y     b 

13-  Theorem.    The  limit  of  the  reciprocal  of  a  variable  equals 
the  reciprocal  of  its  limit. 

With  the  same  notation  as  before,  we  are  to  prove  that 


lim    ^  =— . 
X      a 

We 

know 

that 

I 

x'        "^  ' 

hence 

lim    — x*  =lim  x=a, 

X 

But 

lim 

-x^  =lim  —1.  lim 

.x                 x\ 

=  «^  lim    —  , 

\x\ 

hence 

a  .  lim    -   =^a. 

X] 

hence 

lim    -   =— . 
\x       a 

14.  Theorem.    The  limit  of  any  power  of  a  variable  equals  that 
poiver  of  the  limit  of  the  variable. 

With  the  same  notation  as  before,  we  are  to'prove 

lim  x''=^a'\ 
n  being  any  commensurable  number  either  positive  or  negative, 
integral  or  fractional. 


Theory  of  IvImits.  145 

First.  When  71  is  a  positive  integer.  If  in  Art.  1 1  we  let  j', 
z,  etc.,  each  equal  x  then  b,  c,  etc.,  will  each  equal  a,  and  hence 

\\m(xxx  .    .    .  )=aaa  ,    .    . 
or  lim  x"=a". 

Second.     When  ;/  is  a  positive  fraction,  say    .. 

1 
Let  x'^=^y.  ^        (i) 

then  ^=y'\  .  (2) 

hence  by  Art.  7  a—¥^  (^j 

where  d  is  the  limit  of  v;  hence 

Fromri;  .r^=y,  (^) 

p 
hence  by  Art.  7  \\xi\  x^' —\m\.  y^=^b\  (6) 

p^ 
But  from  ("4;  b^^a'f , 

L        .t 
hence  lim  jf /=«''; 

therefore  the  theorem  is  true  for  any  positive  exponent  whether 
integral  or  fractional. 

Third.  Let  n  be  a  negative  quantity  either  integral  or  frac- 
tional, say  71— —s,  then  jr~'=--;  therefore 

lim  x~'=~=a~\ 
a' 

hence  the  theorem  is  true  for  any  commensurable  exponents. 

INCOMMENSURABLE    POWERS. 

15,  Ii^  Chapter  II  we  have  found  that  whatever  commensur- 
able numbers  are  represented  by  7i  and  r  then 

a"  .a'-^a'*^'-  (a) 

r«" /=«'"-  (b) 

a" -i:-a'^=a"~''  (c) 

but  no  meaning  has  yet  been  given  to  quantities  with  incommen- 
surable indices. 

The  quantity  raised  to  a  power  is  called  the  I^ase.  In  Chapter 
II,  the  base  was  either  positive  or  negative,  but  the  present  dis- 
cussion is  confined  to  the  case  where  the  base  is  positive. 


146  Algebra. 

,    A  power  of  a  given  base  may  have  more  than  one  value,  as  for 

instance,  (^25/^  =  ±5,  but  any  comnieiisurable  power  of  a  base  has 
amouQ;  its  values  one  which  is  positive. 

For  any  integral  power  of  a  positive  base  is  evidently  positive, 
and  ati}'  7'oot  of  a  positive  base  has  among  its  values  one  which  is 
positive,  and  since  any  power  of  this  root  is  positive  so  any  posi- 
tive or  n^gaJdvc  fractio7ial  power  of  a  positive  base  has  among  its 
values  one  which  is  positive,  and  as  a  negative  power  of  a  base  is 
the  reciprocal  of  a  positive  power  of  the  same  base,  any  negative 
fractional  power  of  a  positive  base  has  among  its  values  one  which 
is  po'sitive.  This  positive  value  is  all  that  is  considered  in  the 
present  discussion.  So  that  whenever  we  deal  with  a  quantity 
like  a'  in  the  present  chapter,  both  a  and  a^'  are  positive.  These 
restrictions  must  not  be  lost  sight  of. 

16.  Theorem.  If  x  and  y  are  aiiy  two  coni77iensurable 
numbers  where  y  is  greater  thari  x,  then  a'  is  greater  than  a""  if  a  is 
greater  than  unity ^  and  a^  is  less  thari  a""  if  a  is  less  thaji  U7iity. 

First  Case,     When  a  is  greater  than  unity. 
'■    -^         -  a^ -^a""  =a^-'' 

and  since  _y>jf,  y—x  is  positive,  and  therefore  «'"'  is  greater  than 
unity,  for  a  positive  power  or  root  of  a  quantity  greater  than  unity 
is  itself  greater  than  unity. 

Hence,  a-'  -r-a'  >  i 

hence  a^^a'  . 

Second  Case.     Where  a  is  less  than  unity. 

As  before  a^'  -r-a"  =<2^~' 

and  «^~''<i,  for  a  positive  power  or  root  of  a  quantity  less  than 
unity  is  itself  less  than  unity. 

Therefore  a^'  -r-^^  <C  i 

and  hence  a^'  <«^ . 

Therefore  if  ^  is  greater  than  unity,  the  greater  x  is  the  greater 
is  <2' ,  or  in  other  words,  if  a  is  greater  than  unity,  a""  increases  as 
X  increases,  and  if  ^  is  less  than  unity,  a'  decreases  as  .r  increases. 

17.  Consider  a  quantity  q  intermediate  in  value  between  x 
and  jj',  then  if  a  is  greater  than  unity  a '  <<3:'^  <«-' ,  and  if  a  is  less 


Theory  of  I^imits.  147 

than  unity  «'>«'''> «^'',  so  whether  a  is  greater  or  less  than  unity, 
rt^is  intermediate  between  a*  and  a^ . 

Now  consider  x  and  j' variables,  but  always  commensurable, 
and  let  x  increase  and  y  decrease,  and  suppose  them  lo  approach 
the  same  incommensurable  limit  71,  As  x  and  j/  are  commensur- 
able, a'' and  a^have  definite  meanings,  and  as  x  ahdjj/  approach 
equality,  (one  increasing  and  the  other  decreasing),  «^  and  «-^also 
approach  equality,  or  in  other  words  there  is  some  quantity  be- 
tween ^'^and  «^  from  which  each  of  these  quantities  may  be  made 
to  differ  by  an  amount  as  small  as  we  please. 

But  ^*  and  a''  can  never  become  equal,  since  x  and  y  cannot 
become  equal,  hence  each  of  these  quantities  approaches  the  same 
limit. 

Since  we  have  now  proved  that  both  ^^''and  a^ approach  a  limit, 
as  X  and  y  themselves  approach  a  limit,  we  may  if  we  choose 
neglect  J  and  «^and  fix  our  attention  upon  x  and  a"  remembering, 
however,  that  x  varies  just  as  it  varied  before,  and  hence  just  as 
before  a"-  approaches  a  limit  and  indeed  the  same  limit. 

This  limit  we  will  represent  by  ^" .  Thus  we  have  a  meaning 
for  «"  where  n  is  incommensurable,  viz:  it  is  the  limit  approached 
by  a'   {x  being  commensurable)  as  x  approaches  a  limit  n. 

18.  Extension  of  Formula  {a)  of  Chapter  II  to  Incom- 
mensurable Indices. 

So  long  as  x  and  y  are  commensurable  we  know  that 

a-a'^a""^'  (i) 

Let  X  approach  an  incommensurable  limit  n,  and  y  approach  an 
incommensurable  limit  r. 

Then  x-^y  approaches  a  limit  n-\-r  which  is  usually  incom- 
mensurable, but  may  possibly  be  commensurable.  Also  a"-  ap- 
proaches a  limit  a"  ,  a'  approaches  a  limit  a''  and  a'  ^  approaches  a 
limit  a"+^ 

Now  in  equation  (i)  the  left-hand  member  is  one  variable,  and 
the  right-hand  member  is  another  which  is  equal  to  the  first. 
Hence,  Art.  7,  lim  «*^«-''=lim  a'''^'' . 

But  the  left-hand  member  is  the  product  of  two  variables,  hence 
by  Art.  10,  lim  (a"" a"  )=^\\r\\  a'  lim  a^  or  \im(a'  a^  )=a" a'^ ,  hence 
a"a'^=^a"+''. 


148  Algebra. 

19.  ExTP^NSiON  OP  Formula  (b)  of  Chapter  II  to  In- 
commensurable Indices. 

With  the  same  notation  as  in  the  previous  article  we  have 

{a^)y^a^y  (i) 

hence  Xxva  ( a""  y  =-X\vci  a^^'  (2) 

and  b}^  Art.  7  lim  <a;'^'=<2"''  (3) 

I^et  x—n-\-ii  and  as  x  may  be  made  to  differ  from  its  limit ;/  by 
an  amount  as  small  as  we  please,  21  may  be  made  as  small  as  we 
please,  or  the  limit  of  21  is  zero. 

Now,  by  Art.  17,     (a "/  =  ("«""'">''  =  T^"  «"  /  (4) 

and  because  y  is  commensurable 

(a^^  aJ'  y  =  (a"  y  (a '' y  =  (a^'  y  a''\  (5) 

Substitute  in  (4)  and  we  get 

(a^y=(a'^ya"\  (6) 

hence  lim  ("«">"=  lim  [fa"  >"«"-^],  Art.  7,  (7) 

and  lim  l(a"ya"y']=.\im  (a"y  lim  a"-\  Art.  10       (8) 

therefore  lim  ("«"  y  =lim  (a"  y  lim  a"-\  (g ) 

But  since  J/  approaches  r,  therefore 

lim  (a")^'=(a"y,  (10) 

and  because  //  approaches  o  and  y  approaches  r,  and  hence  uy 
approaches  o,  therefore 

lim  a"-'=a''=  i .  (11) 

Substitute  for  the  right-hand  member  of  (9)  the  values  found  in 
(10)  and  (11)  and  we  get 

lim  (a" y=(a"  y.  (12) 

Substitute  for  the  two  sides  of  equation  (2)  the  values  found  in 
(12)  and  (3)  respectively  and  we  obtain 

(a''y=a"''. 

20.  Extension  of  Formula  (c)  of  Chapter  II,  to  In- 
commensurable Indices. 

With  the  same  notation  as  in  the  two  previous  articles  we  have 

a- -^^•"  =«—-'■  (i) 

Since  lim  x=n  and  lim  j'=r, 

lim  (x—j)^=n—r 
and  lim  a'''=a" ,  lim  a\=a'' 

and  lim  <2 '"-'==«""'. 


Theory  of  Limits. 


149 


From  (i)         lim  (a'  -7-a'  )  =  \\m  ^'"•''by  Art.  7. 

But  lim  (a''  —a'  )=\\m.  a'  -r-lim  a^  =a"  -r-a 
ami  lim  «'~''=a""'', 

therefore  a"  -k-a''  =a"~'' . 


21.     It  is  also  eas}'  to  see  that  where  71  is  incommensurable 

a"b''=(ab)" 


and 


Hence 
But 
and 
hence 

Again 


hence 


and 


hence 


lim 


a' 

lim  a' 

a 

F 

lim  b""     b' 

a   '         a 

" 

X          [b 

a" 

a\" 

b"~ 

b\ 

by  Art.  7. 


For  let  t7  and  b  be  tvvo  bases  and  x  a  variable  which  remains 
commensurable,  but  approaches  an  incommensurable  limit  7t,  then 

a''b'=(ab)' . 
lim  d;"<^=lim  (ab)' 
lim  a''  br'  =lim  a' lim  b'-  =a"  b"  . 
lim  (aby  =-(al)" , 
a"b^=(ab)" . 

b^       Ui 


22.     Examples  of  Limits.    In  the  following  examples  an 

expression  like  ,         \ r  MS  to  be    read:    the   limit  of\—~^\ 

h  'l  a  {a-\-h)  -^    iw^h) 

as   h    approaches  a    as   a  liinit.     The   symbol  ^    stands   for   the 

word  approaches. 

limit  (  (x-\-Ji)-'—x^  \ 


Find 


h  :  o 


T 


Process. 


limit  {(x-k-  /!/— .v"|       limit  f 
>^  ::  ol  S~h  :  o  I 


x^-\-2hx—h^—x''  I 


_  limit  J 
~/^  :  ol 


2X-\-h 


}=.,. 


150  Algebra.  " 

^.    ,   limit  (       mx     ") 
2.     Find      ^     \^    ^ > 

?.     Find  "'"*|?l--:q 
x^  a ( a—x J 

^..  ,  limit  (^r^-/^;'--^^-M 

^  h  ^  o  {  h  \ 

^.    .   limit  ( -r^+i  ) 
5".     Find      ^      \  -  „-     .- 

^       ,,  limit  f.r"— a") 

^.      Prove      ^      -    =na"   '. 

.V  ^  ^  (    .r— ^     I 

23.  lyiMiT  OF  THE  Sum  of  a  Decreasing  Geometrical 
Progression  as  n  Increases.  It  was  noticed  in  Chapter  VIII 
that  if  the  ratio  of  a  geometrical  progression  is  less  than  unity, 
each  term  of  the  series  is  necessarily  less  than  the  one  preceding 
it.     In  this  case  the  series  is  called  a  decreasing  progression. 

In  the  case  of  a  decreasing  geometrical  progression,  it  is  a  little 
better  to  write  the  expression  for  the  sum  of  the  series  in  the  form: 

I  —  r 
Now  if  we  like  we  may  consider  n  a  \'ariable,  and  then  the 
two  sides  of   this  equation   are   two   variables  that   are  always 
equal.     Therefore,  their  limits  are  equal.     Whence  we  may  write 

lim  ^  as  n  increases=lim-i     *      -    —  :as  n  increases. 

I        I  — r     J 

Now  since  r  is  less  than  i,  the  term  ;-"  continually  approaches 
the  limit  o  as  n  increases.  Whence  taking  the  limit  of  the  right 
hand  member  of  the  equation,  we  may  write  : 


,.  .  a 

hm  .V  as  ;/  increases= 


I  — r. 

24.     Examples. 

I.     Find  the  limit  of  the  progression  T-SSSS  +  J  ^"^  ^'  increases. 
Here  ^«=to  and  r=-f^. 

Whence,  lim  .y=    '  ^  ^  =|. 

^~T0 

Therefore,  lim  .3333+ =15. 


Theory  of  Limits.  151 

2.     Find  the  limit  of  the  progression  ir-|-|  +  |  +  TV+  ^^c,  as 
n  increases. 

J.     Find  the  limit  of  .272727+  as  n  increases. 
^.     Find  the  limit  of  .2 792 792 79 -f  as  n  increases. 

5.  Find  the  limit  of  the  sum  of  i~>y  +  TV~A+  ^t<^-'  ^^  ^ 
increases. 

6.  Find  the  limit  of  the  sum  ofV8+\/4+V2-fV  1  + 
as  ;/  increases. 

25.  Theorem.  The  limit  of  the  sum  of  the  series  i-\-r-\-r'-\- 
^-3_j_^.4_j_  ^^^^  ^^  y  decreases  and  n  increases  is  i. 

In  the  equation 

,.  a 

lim  .T=      — 
I— /' 

the  expression         will  of  course  have  different  values  for  different 
I — r 

values  of  r.  Hence  we  may,  if  we  choose,  look  upon  this  expres- 
sion as  a  variable.    But  as  ;-  approaches  o  as  a  limit  the  fraction 

approaches  a  as  a  limit.  Therefore  we  ma}*  say  that  in  a  decreas- 
ing geometrical  progression  as  the  number  of  terms  increases  with- 
out limit,  and  as  the  ratio  approaches  zero  as  a  limit,  the  sum 
approaches  <?  as  a  limit. 

In  particular,  then,  if  a=-\  and  if  the  number  of  terms  in- 
creases without  limit,  and  the  ratio  approaches  zero  as  a  limit,  the 
series 

I +  /'+/"+  ... 
approaches  i  as  a  limit. 

26.  Theorem.      The  limit  of  the  series 

as  the  number  of  terms  increases  loithout  limit  and  as  x  approaches 
zero,  is  A^. 

Take  the  series  first  without  the  A,,,  and  suppose  K  to  be  a 
positive  quantity  numerically  equal  to  the  greatest  of  the  co- 
efficients A,,  A,^,  A^,  .    .    .    Then 

A -x--f  A  A"+A.r^-f  .    .    .  nunierically<K('A-+.i"-fA-^-h  .    .  ) 


152  AlvGEBRA. 

By  Art.   25  the  limit  of  i -\- x -\- x' -\-  .    .    .  ,  as  the  number  of 
terms  increa.ses  and  as  x  approaches  zero,  equals  i.    Therefore, 

lim  (x-j-x''-^x^-{-  .    .    .  )=o 
and  consequently 

lim  K(x+x'^x'-\-  .  .  .  )=o. 
That  is  to  say,  the  right  member  of  the  inequality  above  can 
be  made  as  near  zero  as  we  please.  Therefore  since  the  left  mem- 
ber of  the  inequality  is  always  numerically  less  than  the  right 
member,  the  left  member  can  be  made  to  approach  zero  as  near  as 
we  please.     Hence, 

lim  (Ax-\-Ax^'  +  Ax^'-\-  .    .    .  j=o. 
That  is         lim  ('A,-f-A^.r  +  A^.v=-f  Ax^-f  .    .    .  ;  =  A, 


CHAPTER  XII. 

UNDETERM I  NED    COEFFICENTS. 

x'' a^  \ 

1.  We  know  that—      -=^x-\-a,  and  if  we  integralize  this  we 

obtain  an  equation  of  the  second  degree,  but  an  equation  of  a 
different  kind  from  those  treated  in  Chapters  IV  and  V,  for  the 
equations  previously  treated  under  the  name  quadratics  were 
shown  in  Chapter  V,  Art.  9  to  have  two  roots,  and  only  two;  that 
is,  it  was  shown  that  there  were  two  and  only  two  values  of  the 
unknown  quantity  which  would  satisfy  the  equation  ;  but  here 
we  have  an  equation  of  the  vSecond  degree  which  can  be  satisfied 
by  any  value  whatever  of  x. 

The  reason  is  that  when  the  equation  is  in  the  integral  form  we 
have  exactly  the  same  function  of  x  on  each  side  of  the  sign  of 
equality. 

2.  Theorem.  If  two  functions  of  x  of  the  n  th  degree, 
A„H-A^.r+  .  .  .  -\-A„x"  and  B^+Bx-\-  .  .  .  -i-B^x" ,  are  e^ua/ 
for  every  value  of  x,  then  the  coefficients  of  like  powers  of  x  07i  the  tzvo 
sides  of  the  sign  of  equality  are  equal  each  to  each. 

If  the  two  functions  are  equal  for  every  value  of  x,  we  have 
A,+Aa-+  .    .    .  -f-A„x"=B„+B,.r+  .    .    .  +B„a-",       (y) 
and  since  this  equation  is  true  for  any  value  of  x,  we  may  con- 
sider X  as  a  variable,  varying  in  any  way  we  please. 

Then  if  we  consider  x  to  approach  a  limit,  each  side  of  the 
equation  is  a  variable  which  approaches  a  limit,  and  we  have  two 
variables  which  are  always  equal,  aiid  each  approaches  a  limit, 
hence  by  Chapter  XI,  Art.  7  the  limits  are  equal.  Suppose  x  to 
approach  zero  as  a  limit  then 

limit  of  A„+A  x+  .    .    .  +A„ji-"=A„ 
and  limit  of  B^  +  B^-r-h  .   -    .  +B,a-'=B^, 

hence  A^=B„  by  Chap.  XI,  Art.  7.  (2) 

Subtracting  A^  from  the  left  side  and  B^  from  the  right  side  of 
(\)  we  get 

Kx^-Kx^^-  .    .    .  +A„-t-"=B,.r+B^a-+  .    .    .  H-B„,v'     (i) 
Divide  (^3  j  by  x  and  we  have 

A,4-A,-r4-  .    .    .  +A„A-"-==B,  +  B_..v+  .    .    .  +B.,.v"-'     (^^) 

A-19 


154  AlvGEBRA. 

Again  let  x  approach  zero  as  a  limit,  then 

hmit  of  A,  +  A^_r-f  .    .    .  4-A„ji-"-'=A,, 
limit  of  B,  +  B^j«;+  .    .    .  H-B„.r"-'  =  B,, 
therefore  A^=B,  by  Chap.  XI,  Art.  7.  ^5; 

Subtracting  A^  from  the  left  vSide  and  B^  from  the  right  side   of 
(4)  we  get 

Ax-\-Ax^~-\-  .    .    .  +A„x"-  =  Bx+Ba-H-  .    .    .  +B„A-"- f6; 
Divide  (6)  by  x  and  we  have 

A^+A^jt-^  .    .    .  4-A,X'-^=E^+B3X+  .    .    .  -f-B,,.v"-^     (7) 
Then  in  same  way  as  in  the  two  preceding  instances  it  follows 
that  A^=B^ 

and  bv  continuing  the  process  we  get 

etc. 
Therefore  if  the  two  functions  are  equal  for  all  values  of  x\  the 
coefficients  of  like  powers  of  x  in  the  two  functions  are  equal 
each  to  each. 

3.  Equations  of  the  kind  just  considered,  which  are  satisfied 
by  a?ij'  value  of  x  are  often  called  Identical  equations,  while  those 
with  which  algebra  has  most  to  do,  those  satisfied  by  particular 
values  of  x  equal  in  number  to  the  degree  of  the  equation,  are 
often  called  Conditional  equations. 

4.  Definitions.  A  Series  is  a  succession  of  terms  each  of 
which  is  derived  from  one  or  more  of  the  preceding  ones  by  a 
fixed  law.  An  Infiyiite  Series  is  one  in  which  the  number  of  terms 
is  unlimited. 

An  infinite  series  is  Co7ivergent  if  the  sum  of  the  first  n  terms 
approaches  a  limit  as  n  increases  without  limit. 

An  infinite  series  is  Divergent  if  the  sum  of  the  first  n  terms 
does  not  approach  a  limit  as  n  increases  without  limit. 

The  series  \ -\- x -\- x"" -\-  .  .  .is  convergent  if  x  is  less  than  unity, 
but  divergent  if  x  is  equal  to  or  greater  than  unity. 

5.  Theorem.  If  for  every  value  of  x  which  makes  each  of  the 
two  series  A  -\-A  x-\-  .    .    .  and  B  -\-B  x-\-  .    .    .  coyivergent  these 


Undetermined  Coefficients.  155 

two  series  approach  the  SAME  limit  as  the  number  0/  terms  increases 
7vithout  limit,  then  the  coefficients  of  like  powers  of  x  in  the  two  series 
are  equal  each  to  each. 

Since  we  are  dealing  with  the  limit  of  convergent  series  as  the 
number  of  terms  increases  without  limit,  we  know  that  by  taking 
a  sufficient  number  of  terms  the  sum  of  the  terms  taken  may  be 
made  to  differ  from  the  limit  of  the  sum  by  an  amount  as  small  as 
we  please. 

Let  us  then  write 
limit  (A„-f  A  A-  +  A  A-+  .    .    .  +A„_,;r"-'+  .    .    .  ) 

=  A^,+A,-^-^-Ax^+  .  .  .  +A,,_i,r"-'  +  R,;i-", 
where  R^Jt"  is  of  course  the  difference  between  the  limit oi  the  sum 
as  the  number  of  terms  increases  without  limit  and  the  actual  sum 
of  the  first  n  terms. 

A^,  Aj,  .  .  .  A„_i  are  each  constant,  but  R^  is  not  constant, 
for  if  it  were  the  series  would  terminate.  In  fact  ^x"  approaches 
zero  as  n  increases,  for  if  it  did  not  the  series  would  not  be 
convergent.  An  inspection  of  the  series  shows  that  every  term 
after  the  first  contains  x,  every  term  after  the  second  contains  x', 
ever>'  term  after  the  third  contains  x^,  and  so  on  ;  hence  every 
term  after  the  n  th  will  contain  the  factor  x'\  and  so  it  is  natural 
to  assume  the  remainder  after  71  terms  are  written  to  be  of  the 
form  R^jt". 

Instead  of  writing  limit  of  A^-t-A^.r-f-  ...  as  the  number  of 
terms  increases  without  limit  we  write 

A„+A  A-+A  x=+  .    .    .  A„_i-t"-'  +  Rx" 
and  in  the  same  way  write 

B,,+Bx+BX+  .    .    .  +B„_iX"--hR,-^t-" 
instead  of  writing  limit  of  B^ -f  B^.v  +  B.x^  .    .    .as  the  number  of 
terms  increases  without  limit. 

Using  this  notation  we  may  write 
A,,+A  A-+  .    .    .  -f  A„_iX"-'-|-Rx" 

=  B„+Bx+  .    ,    .  +B„_i.v"-'-hRx".   (i) 

If  now  we  consider  .v  as  a  variable  approaching  zero  we  have 
here  two  variables  which  are  always  equal,  and  therefore  by 
Chapter  IX,  Art.  7,  their  limits  are  equal.  By  Chapter  IX, 
Art.  26,  the  limit  of  left-hand  member  equals  A^,  and  the  limit 
of  the  right-hand  member  equals  B^*  hence  A^^=B^. 


156  Algebra. 

Subtract  A,^  from  the  left-hand  member  and  B_^  from  the  right- 
hand  member  of  (^  i  J  and  we  get 

A,-r+A,j»:^+  .    .    .  +Rx"==Bx+BX-+  .    •    •  4-RX'     (2) 
Divide  both  members  of  (2)  by  x  and  we  get 
A^-\-A'x+  .    .  +A,_iJt-"-=+Rjtr""' 

=  B^  +  B^jt--f  .    .    .  +RX'-'.    (3) 
As  before,  we  have  two  variables  always  equal,  hence  theii 
limits  are  equal. 

But  as  X  approaches  zero  the  limit  of  the  right-hand  member 
equals  A^  and  the  limit  of  the  left-hand  member  equals  B^. 
Hence,  by  Chapter  XI,  Art.  7, 

A=B,. 
Repeating  the  reasoning,  we  may  show  successively  that 

A^=B^, 

a=b;, 

etc. 

6.  The  theorem  of  the  last  article  will  enable  us  to  change  the 
form  of  a  function. 

The  method  of  doing  this  consists  in  assuming  a  function  of 
the  required  form  with  unknown  coefficients  and  then  determin- 
ing the  coefficients  so  that  the  function  assumed  shall  be  identical 
with  the  function  proposed.  The  unknown  coefficients  are  deter- 
mined b}^  placing  the  proposed  function  equal  to  the  assumed 
function,  reducing  to  the  rational  integral  form,  and  equating  the 
coefficients  of  like  powers  of  the  variables  on  the  two  sides  of  the 
equation. 

If  the  proposed  function  can  be  placed  in  the  assumed  form  it 
will  be  found  that  there  are  as  many  independent  compatible 
equations  as  there  are  unknown  quantities  to  determine. 

7 .  Definition.  A  function  is  said  to  be  Developed  or  Ex- 
panded when  it  is  expressed  in  the  form  of  a  series,  the  sum  of 
whose  terms  when  the  number  of  terms  of  the  series  is  limited, 
and  the  limit  of  the  sum  when  the  number  of  terms  is  unlimited, 
equals  the  given  function. 

8.  The  development  of  functions  is  one  of  the  most  common 
applications  of  the  method  described  in  Article  6.     The  process 


Undetermined  Coefficients.  157 

is  usually  referred  to  as  the  method  of  undetermined  coefficients. 
We  will  illustrate  the  method  by  working  an  example. 

I^et  us  develope  the  fraction        -. 
Assume 

_^^_=A„-f  AA-+A^.r=-f  .    .    .  -f  A,_i.r"-i4-Rjt-".  (\) 

Multiply  both  sides  of  (^ij  by  \—x  and  we  get 
i=A +rA,-AJ.r+rA^-A.X+  .    .    . 

+  rA_i-A„_2>-"-i-t-rR-A„_Jjt-"-R;i-"-i. 
We  see  that  the  left  hand  member  contains  no  power  of  x  ex- 
cept the  zero  power,  or,  in   other  words,  in  the  left  hand  m^iber, 
the  coefficients  of  the  various  powers  of  x  except  the  zero  power 
are  each  zero.     Hence  equating  coefficients  we  get 
A=i 

A-A=o.-.  A=A, 

A-A=o.-.  A^=A^, 

A3— A^=o  .-.  A3=A^, 

etc.,  etc. 

From  these  equations  the  law  of  the  series  is  so  evident  that 

we  can  wTite  as  many  more  equations  as  we  please  without  further 

calculation. 

We  thus  see  from  the  second  column  of  equations  that  each 
coefficient  equals  the  preceding  one,  and  as  the  coefficient  of  x", 
or  the  absolute  term,  equals  i;  therefore  each  of  the  other  coeffi- 
cients equals  i .     Hence  we  obtain 

— --  =  i+-r+-r^+-v^  +  -r^+  .    .    . 
I — X 

As  we  usually  determine  only  a  few  of  the  coefficients,  and  then 

discover  if  we  can  the  law'  of  the   series,   so  it  is  usual  in  the 

assumed  series  with  undetermined  coefficients  to  write  only  a  few 

terms  and  indicate  the  others  including  the  remainder  by  dots 

thus: 

~-   =A,  +  AA-+A.r^-fA-r^+  .    .    . 
I— .r     .  '  -  ^ 

Instead  of  using  the  method  of  undetermined  coefficients  we 
might  have  proceeded  by  ordinary  long  division  as  follows: 


158  Algebra. 


A-  I         ;i+-v  +  a---f  A-  +  .r^ 
1—  X 

X 

X—X' 

Jtr* 
X^—X^ 

x^ 

x^—x* 
x\ 


Here  as  before  we  obtain 

I— .r 
This  series  on  the  right  side  of  the  sign  of  equality  is  conver- 
gent if  x<,i,  but  not  otherwise,  and  therefore  this  series  cannot 

be  called  the  development  of  unless  .v   is  less   than  unity. 

See  Art.  7.     When  .r  is  equal  to  or  greater  than  unity  the  fraction 

^  cannot  be  developed. 

9.  Examples.  Develope  the  following  fractions  both  by  the 
method  of  undetermined  coefficients  and  by  actual  division,  and 
in  each  case  discover  the  law  of  the  series. 

Also  in  each  case  state  for  what  values  of  x  the  series  is  a  true 
development. 

I  i-^x 

\-\-x  \—x-\-x- 

Ijf-V  I  +  2.V-}- 3JI:' 

I — X'  l—2X-\-T,X^' 

^  «  ^,  1-2X±SX^^    . 

4.     '"^3-^'.  o.     _5£+7^. 


5-  7      ^-  ^o. 


SX"" — T,X 


l  —  2X-\-  TfX"  3  —  4.x''  4-  2X^ 

Compare  the  laws  of  the  series  in  the  developments  of  the  frac- 
tions in  examples  i  and  4  ;  also  compare  the  laws  of  the  series 
in  examples  5  and  7  ;  also  in  examples  9  and  10. 


Undetermined  Coefficients.  159 

Query  :  What  controls  the  law  of  the  series  in  the  develop- 
ment of  a  fraction  ? 

Query  :  How  does  the  numerator  affect  the  development  of  a 
fraction  in  the  fonn  of  a  series  ? 

Query  :  What  would  the  results  to  examples  7  and  8  suggest 
about  the  development  of  fractions  which  are  reciprocals  ? 

10.  It  sometimes  happens  when  we  try  to  develope  a  fraction 
by  the  method  explained  that  some  of  the  equations  are  absurd 
or  contradict  one  another. 

The  reason  of  this  is  because  the  fraction  cannot  be  developed 
into  a  series  of  the  form  assumed.     Thus  if  we  try  to  develope 

I 

we  assume 

x—x- 

X  —  X-  0.2  3  4 

Multiply  by  x—x"  and  we  get 

i=AXA  AJx^-|-fA^AjA-^+  ... 
hence  1=0, 

A=o, 

A=o, 

A^===o, 

etc. 

But  the  first  of  these  equations  is  false,  so  we  consider  that 

the   function  cannot  be   developed  into  a  series  of  the  assumed 

form. 

But  we  note  the   denominator  of  the  given  fraction  contains  a 

factor  A-,  and  that  hence  the  fraction  proposed  equals      •  ,  the 

second  factor  of  which  has  already  been  developed. 

From  this  we  would  infer  that  the  development  of  ^  could 

be  obtained  from  the  development  of  by  dividing  every  term 


in  that  development  b}-  .v. 


i6o  A1.GEBRA. 

Hence  — ^=x~^  -j-i-\-x-{-x^-{-x^^-h.   .    . 

x—x^ 

and  we  would  obtain  this  very  result  if  should  assume 

^:^,^Ax-^-hA^-\-A^x-\-Ax-+.    .    . 

or  in  other  words  if  we  de^in  the  assumed  series  with  a  term  con- 
taining x~^  instead  of  beginning  with  an  absolute  term.  If  the 
fraction  we  wish  to  develope  is  in  its  lowest  terms,  and  if  the 
lowest  power  of  x  that  appears  in  the  denominator  is  the  rth 
power  then  we  must  begin  our  assumed  series  with  a  term  con- 
taining x~''. 

This  is  a  safe  rule  whether  the  fraction  is  in  its  lowest  terms  or 
not,  but  it  i's  not  always  necessary  when  the  fraction  is  not  in  its 
lowest  terms. 

In  any  case  when  we  form  an  equation  by  putting  a  given  frac- 
tion on  the  left  and  an  assumed  series  on  the  right  side  of  the  sign 
of  equality,  the  assumed  series  must  begin  with  such  a  power  of 
X  that  when  the  equation  is  integralized  the  lowest  power  of  x  on 
the  right  side  of  the  equation  will  be  as  low  as  the  lowest  power 
on  the  left  side. 

11.  Examples. 

\-\-x-\-x^--\-x^ 

1.  Develope , , — . 

X^-\-2X^ 
^  ,  X+'IX'^ 

2.  Develope  — —^,. 

^     X^-\-  2X^ 

12.  Not  only  fractions  but  some  irrational  expressions  may  be 
developed  by  the  method  of  undetermined  coefficients. 

Let  us  develope  Vi — ■^• 
Assume 

^/Y^x=A^-]^Ax-\-Ax'  +  Ax^  +  Ax'-{-A_x^-\-  .    .    . 
Square  each  side  and  we  get 

i-x=A-f2AA^jt-4-(2AA+A;K+(2A>.3+2AA>^ 

+  (2A  A^+2A  A3-f  A;>-^4-(2A„A^  +  2A  A^+2A^A3>5+  .    . 

Equating  coefficients  of  like  powers  of  x  we  get 


Undetermined  Coefficients.  i6f 

A=i, 
2A^A,=  — I, 

2A^A,+  2A  A^=o, 

2A^a^+2AA,+a;=o, 

2AA^+2A  A^4-2A^A^=o, 

2AA+2A,a^4-2AA^+a;=o, 

etc. 
From  these  we  get 

A=i, 

_     2  A,  A, 


2A„      • 

A  _     ^A,A,+2A,A.+A/ 

etc. 
From  these  the  law  of  the  series  can  be  seen. 
Taking  these  equations  in  order,  we  find  the  numerical  v^ahie 
of  the  undetermined  coefficients  to  be  as  follows  : 

Ao=i»     A^=— J,     A^=— -|-,     A^=— jV,     A^=  — i|-g-, 

A    — 7  A 21 

^5—     TTe'     ^6—  r(5"2T- 

Making  these  substitutions  in  the  assumed  development,  we 
obtain 

. _        X     jr      x^      ^x^  yx^      2i-V^ 

2      8      16     128  256     1024 

13.  Examples. 

I  ^ 

I.     Develope       1+2  x—     • 


2.     Develope  \/x-j-x'\ 

J.     Develope  (i-\-x)^. 
A-20 


1 62  Algebra. 

14.  It  is  interesting  to  note  that  the  development  of  an  irra- 
tional expression  7?iay  turn  out  to  be  a  series  of  a  limited  number 
of  terms. 


Suppose,  for  example,  we  wish  to  develope  s/  i  —  2x-\-x^  and 
do  not  recognize  that  i  —  2x-\-x^  is  a  perfect  vSquare,  then  assume 
as  before 


s/  i  —  2x-]-x^=A^^-\-Kx-\-Kx'+  .    .    . 
Square  both  sides  and  we  get 
I  — 2.:»;— jL'^=A^,+  2A^Aa'-|-(A,^+2A^AJx^ 

■    +(2A^A^-}-2A  A^>-^-f  .... 
Bquating  coefficients  of  like  powers  of  x  and  we  get 

A„=i, 
2A^Aj=  — 2  .'.  A^=  — I, 
A;-f2AA=i  .  .  A  =o, 
2A^A^-f  2A  A^=o  .-.  A^=o, 
A;H-2A  A^+2A  A  =o  .-.  A^=o, 
etc.,  etc., 

and  each  of  the  subsequent  coefficients  will  turn  out  to  be  zero, 
hence  we  ^et 

S/  l  —  2X-\-X''^==l—X. 

15.  In  developing  irrational  expressions  it  sometimes  happens 
that  we  should  begi7i  our  assumed  development  with  some  negative 
power  of  X. 

An  inspection  of  the  p:op33ed  example  will  show  with  w^hat 
power  of  jr  the  development  should  begin  ;  for  the  assumed  series 
must  be  such  that,  when  the  equation  obtained  by  putting  the 
given  function  equal  to  the  assumed  series  is  reduced  to  the 
rational  integral  form,  then  the  lowest  power  of  x  on  the  side 
which  contains  the  undetermined  coefficients  must  be  as  low  as 
the  lowest  power  on  the  other  side  of  the  equation. 

Thus,  to  develope  ^M+    2  we  would  begin  the  assumed  series 

with  a  term  containing  x~\  for  when  this  is  squared  the  lowest 
power  of  X  is  x~^  and  when  both  sides  are  multiplied  by  x"" 
to  reduce  to  the  integral  form  then  the  series  on  the  right  side  of 
the  equation  will  begin  with  an  absolute  term  as  it  should. 


Undetermined  Coefficients.  163 

16.  If^we  wish  to  develope  the  algebraic  sum  of  two  or  more 
radicals  it  is  best  to  develope  each  one  by  itself  and  then  find  the 
algebraic  sum  of  the  results. 

17.  Examples. 

/.     Develope     1 1  —  \. 

2.     Develope  Ax-\-  --+  sj  i  -f-i". 


' .     Develope  %/  i  -f-  4A- + 6-r"  -|-  6-r  ^ + 5^1-  *  -}-  2-r  ^ + x"". 


4.     Develope     |.r^4- 2. V -1-3  +  ^  +— . 


X     x^ 


CHAPTER  XIII. 

DERIVATIVES. 

I.  Notation.  A  definition  of  a  function  of  a  quantity  was 
given  in  I,  Art.  i.  To  designate  a  function  of  x  we  use  the 
notation /fxj. 

A  function  of  a  quantity  is  denoted  by  writing  the  quantity  in 
a  parenthesis  and. writing  the  letter  f  or  F  or  some  other  func- 
tional symbol  before  the  parenthesis,     e.  g-. 

f(x),  F(x),  F/x)  denote  functions  of  ;i', 

f(y),  F(y),fjy)  denote  functions  of  r, 

f(x-\-h),  F(x-\-h),f'(x-\-h)  denote  functions  oix-{-k, 

f(a),  F(a),f„(a)  denote  functions  of  «. 

The  student  must  be  careful  not  to  look  upon  the  expression 
f(x)  as  meaning  /times  x.  The  symbol  /as  used  here  is  not  a 
quantity  at  all,  but  simply  an  abbreviation  for  the  words 
ftindion  of. 

It  frequentl}'  happens  that  in  the  same  discussion  we  wish  to 
refer  to  different  functions  of  x,  in  which  case  we  use  different 
functional  symbols,  as  F( x ) ,  f( x ) ,  fj x ) ,  /„( x ) ,  F„(x),  etc. 

It  also  frequently  happens  that  in  the  same  discussion  we  wish 
to  refer  to  the  same  function  of  different  quantities,  in  which  case 
we  use  the  same  functional  symbol  before  the  parenthesis  but  dif- 
ferent quantities  within  the  parenthesis,  e.  g.  \i  f(x)  denotes 
jtr^+i  then  f(a)  denotes  a^-\-\,  f(.z)  denotes  .J^-f  i,  etc.,  and  if 
F(x)  denotes  /v/-^4-3  then  F(y)  denotes  Vj'4-3,  F(x+/i)  do:- 
notes  V -^' 4- // +  3 , -etc . 

A  function  of  two  quantities  is  any  expression  in  which  both  of 
the  quantities  appear. 

If  we  have  to  deal  with  a  function  of  two  quantities,  say  x  and 
y,  we  use  the  notation /(^jf,  y)  or  F(x,  y),  and  if,  in  the  same  dis- 
cussion, we  wish  to  speak  of  two  or  more  functions  of  x  and  y, 
different  functional  symbols  are  used,  a.s//x',  y),  f,(x,  y),  etc. 


Derivatives.  165 

2.  In  such  an  expression  as  /(^.r,  j/^  the  two  quantities  x  and 
y   are   entirely  unrestricted   in  value  and   independent  of  each 

other  ;  but  if  we  have  an  equation  like  f(x,  y)=-o,  then  x  and  y 
are  to  some  extent  restricted  ;  any  value  may  indeed  be  given  to 
07i€  of  the  quantities  but  then  the  equation  fixes  the  value  of  the 
othei\  or  in  other  words,  either  one  of  the  quantities  x  or  y  de- 
pends upon  the  other  one;  e.  g.,  \i f(x,y)  stands  for  x—y-\-2 
then  when  this  is  not  put  eaual  to  anything  there  is  no  relation 
between  .v  and  y.  We  may  let  ^  =  3  and  j'=5  or  7  or  10  or  any 
other  number.  But  if  we  put  this  same  function  equal  to  zero, 
then  there  is  some  relation  between  x  and  y  and  they  are  to  some 
extent  restricted  in  value.  We  may  let  ^'=3,  but  then  ^r=5  and 
nothing  but  5. 

3.  If  the  equation  F(x,  y)=^o  can  be  solved  for  j',  we  can  ex- 
press y  in  terms  of  x,  or  y  can  be  determined  as  a  function  of  x. 
If  we  thus  determirue  J  we  havey=/(x). 

In  this  equation,  y=/(x),  we  may  look  upon  x  as  a  variable, 
and  of  course  if  x  varies  r  will  also  var\'.  We  may  consider  x  to 
vary  in  any  way  we  please,  but  then  the  equation  determines  the 
way  in  which  r  varies.  For  the  reason  just  stated  x  is  called  the 
independent  variable,  and  r,  which  is  a  function  oi  x,  is  called  the 
dependen  t  i  'a  Ha  ble. 

4.  In  the  equation  y=f(x)  if  a  value  be  given  to  .v  then  r  will 
have  some  corresponding  value,  and  if  .r  be  given  another  value 
different  from  the  first  one  then  y  will  have  some  value  different 
from  the  one  //  had  at  first.  Moreover,  the  amount  by  which  y 
thus  changes  in  value  will  depend  in  some  way  upon  the  amount 
by  which  .r  changes,  or  in  other  words,  there  is  some  relation 
connecting  the  change  in  value  of  y  with  the  change  in  value  of 
X.  This  relation  we  will  examine,  and  it  will  be  found  to  be  a 
very  important  relation  in  all  that  follows. 

5.  Suppose  f(x)  to  stand  for  2x-\-\,  then  putting  this  equal  to 
r  we  have 


1 66  Algebra. 

Let  us  now  give  to  .r  a  series  of  values,  sav  the  successive  in- 
tegers from  I  to  lo,  and  in  each  case  compute  the  corresponding 
vahie  of  r.     The  results  may  be  expressed  in  the  form 
r  I  6     8     lo     12     14     16     18     20     22     24 
A-  I  I     2       3       4       5       6       7       8       9     10 
where  any  number  in  the  lower  line  is  one  of  the  values  of  .v  and 
the  number  immediately  above  it  is  the  corresponding  value  of  j'. 
If  ji:=   2  the  corresponding  value  of  r  is    8, 
and  if  .v—  10  the  corresponding  value  of  j'  is  24, 
and  if  X  be  considered  to  increase  from  2  to  10  then  at  the  same 
time  y  will  increase  from  8  to  24,  or,  starting  at  -v=2,  if  x  in- 
crease by  8,  J'  will  increase  by  16,  or  if  the  increase  of  .r  is  8,  the 
corresponding  increase  of  r  is  16. 

Still  starting  at  .v=2,  let  us  increase  x  b}^  various  amounts  and 
determine  the  corresponding  increase  of  r. 
The  results  may  be  arranged  in  the  form 

increase  of  J'  |  16     14     12     10     8     6     4     2 

increase  ofx|8       7       6       54321 

We  might  have  started  with  some  other  value  of  x  than  2  and 

have    obtained    similar   results.     In  every  obsen^ed  case  we  .see 

that  the  increase  of  j  is  just  twice  the  increase  of  .v,  or  in  ever>'' 

observed  case 

increase  of  v 
increase  of  x 
It  is  easy  to  see  that  this  is  necessarily  the  case  whatever  the 
value  of  .V  with  which  w^e  start  and  whatever  the  amount  by 
which  X  is  increased,  for  if  x  increases  by  any  amount,  2X  will 
increase  by  just  twice  that  amount  and  the  change  in  the  value 
of  X  does  not  affect  the  4,  therefore  2.1-1-4,  or  y,  will  increase 
twice  as  much  as  .r  increases,  or 

increase  of  v 
increase  01  x 

6.  Notation.  In  what  follows  we  deal  largely  with  equa- 
tions formed  by  putting  r  equal  to  a  function  oi  x,  and  as  we  will 
make  extensive  use  of  the  increase  in  the  value  of  .v  and  the  cor- 


DerivativEvS.  167 

responding  increase  in  the  value  of  v  it  is  well  to  have  a  conven- 
ient notation  by  which  these  amounts  of  increase  are  denoted.  So 
in  future  we  will  use  Jx  to  denote  the  increase  in  the  value  of  x 
and  Jv  to  denote  the  corresponding  increase  in  the  value  of  r. 

In  this  notation  the  fraction  at  the  end  of  Art.  5  would  be 
written 

The  student  is  cautioned  not  to  think  of  J.v  as  being  J  times 
X,  for  the  symbol  J  as  here  used  does  not  stand  for  a  quantity  at 
all,  but  simply  for  the  words  increase  of. 

7,  Let  us  now  consider  the  equation 

y=^x'-\-\. 
In  this  equation  giv^e  x  the  successive  integer  values  from  —3 
to  7  and   compute  the  corresponding  values  of  y.     We  may  ar- 
range the  results  as  in  Art.  5, 

y_  I  iQ  5  2  I  2  5  10  17  26  37  50 
A-  I  —3  —2  —I  o  I  2  3  4  5  6  7 
If  ;t:=i  the  corrCvSponding  value  of  r  is  2,  and  if -i-=7  the  cor- 
responding value  of  J  is  50,  and  if  x  be  supposed  to  increase 
from  I  to  7,  at  the  same  time  y  will  increase  from  2  to  50,  or 
starting  at  .r=i,  if  x  increase  by  6  then  y  will  increase  by  48,  or 
when  J-v=6,  ^y=\%. 

Still  starting  at  x=  i ,  let  us  give  to  ^x  various  values  and  de- 
termine the  corresponding  values  of  ^y. 
The  results  may  be  arranged  in  the  form 

jj_|  48     35     24     15     8     3 

^x\   'b     5     4     321- 
ji/ 

Here  we  have  a  case  where  the  ratio  ~  is  not  always  the  same 

as  it  was  in  Art.  5,  but  at  one  time  it  is  Y"*  o^"  ^.  ^^  another  time 
it  is  -V-,  or  7,  etc. 

As  can  be  seen  by  the  above  scheme,  the  fraction  -j-  takes  suc- 
cessively the  values  8,  7,  6,  5,  4,  3  as  J.v  takes  ;the  successive 
values  6,  5,  4,  3,  2,  i. 

We  now  give  to  x  values  intermediate  between  i  and  2  and 


1 68  Algebra. 

compute   the   corresponding  values  of   r.     The  results  may  be 
arranged  in  the  form 

y   I  2   2.000020000I   2.0002000I   2.00200I   2.02CI   2.21 
X   I  I   I.OOOOI  I.GGOI  •       I.OOI       I.OI      I.T 

As  before,  let  us  start  at  x=^  i  and  give  to  Jx  various  fractional 
values  and  determine  the  corresponding  values  of  Jr. 

The  results  may  be  arranged  as  before  in  the  form 

Jj/  I  .21     .020 1     .002001     .00020001     .0000200001 
J-T  I  .1       .01  .001  .0001  .00001 

An  examination  of  this  scheme  shows  that 

J  r     .21 
when  J.r=.i,  then  -~-  =  —  =  2.1 
Jjt-      .  I 

1.        <  ^x,       -^y     -^^^oi 

when  J.r=.oi,  then  -7-= =2.01 

Jjt       .01 

.       Jr     .002001 
when  Jjf=.ooi,  then  -^=  =2.001 

Jjf        .001 

,  ,  Jy     .00020001 

when  Ja"=.oooi,  then  -^=  =  2.0001 

Jx         .0001 

,  ,  Ay     .0000200001 

when  Jx=.ooooi,  then  -^^=: =2.00001 

Jx  .00001 

From  the  first  part  of  the  Article  it  appears  that  ~f-_  is  a  var- 
iable, and  from  what  we  have  just  obtained  it  further  appears 

that  as  J.r  is  taken  smaller  and  smaller  the  fraction  -{-  approaches 

J.r 

nearer  and  nearer  the  value  2,  or  in  other  words,  it  appears  that 

the  fraction  -^  approaches  2,  /.  c.  2  times  i,  as  -x  approaches  zero. 

In  obtaining  the  result  it  is  to  be  noticed  that  we  consider  x  to  in- 

crease/r^w  the  value  i,  but  if  we  let  x  increase  b}^  various  amounts, 

beginning  to  count  the  increase  in  x  from  the  value  2,  reasoning 

exactly  as  we  have  just  done  would  lead  to  the  conclusion  that 

Ay 

-y^  approaches  \,  i.  e.  2  times  2,  as  Ax  approaches  zero. 

Again,  if  we  begin  to  count  the  increase  in  x  from  the  value  3 

we  would  be  led  to  the  conclusion  that  -~  approaches  6,  /.  e.  2 

times  3,  as  Ax  approaches  zero. 


Derivatives.  169 

8.  In  general,  if  a  be  taken  as  the  value  of  x  from  which  we  be- 
gin to  count  the  increase  of  Jt:  we  would  judge  from  analogy  that  the 

Ji/ 
fraction—  approaches  2a  as  J-r  approaches  zero,  or,  using  the 

notation  of  Chapter  XI,  Art,  22, 

limit    (Jy) 

This  we  will  now  prove. 

Since  y=x^-{-i,  (i) 

whatever  value  be  assigned  to  x  the  equation  will  enable  us  to 
compute  the  corresponding  value  of  y. 

First,  let  x=^a  and  represent  the  corresponding  value  of  jk  by 
b,  then  b^^a'-^-i.  (2) 

Now  let  x=a-\-Jx 

and  represent  the  corrCvSponding  value  of^by  b-\-^y,  then  from 
equation  (i)  we  get 

^4-4y=r^  +  J-r/+i,  (3) 

or  simplifying, 

b-\-Sy=a'^-2aJx-\-(Jxf^\.  T^) 

Subtract  (2)  from  (\)  and  we  get 

Ay^2a^x-V(^x)\  (5) 

Divide  (^5 J  by  J.r  and  we  obtain 

^=2^  +  J;r.  (6) 

As  Jji"  varies,  of  course  the  the  two  sides  of  equation  (d)  are 
variables,  and,  indeed,  they  are  two  variables  that  are  always 
equal,  and  as  J.r  approaches  zero  these  two  variables  each  ap- 
proach a  limit. 

Hence  by  Chapter  XI,  Art.  7,  their  limits  must  be  equal. 

Therefore  }^^^ ^  \  f^  =  2a. 

Ay 
9.  Definition.  The  value  of  the  fraction  -f-  when  that  frac- 

.       Jr 
tion  is  constant,  or  the  limit  of  the  fraction  -~  as  Jx  approaches 

zero  when  that  fraction  is  a  variable,  is  called  the  Derivative  of  y 

tvith  respect  to  x,  and  is  represented  by  the  notation  D,j',  where  j' 

is  a  function  of  x. 
A— 21 


170  Algebra. 

10.  The  general  method  of  finding  the  derivative  oiy  with  re- 
spect to  X  is  that  used  in  Art.  8,  viz :  give  to  x  some  value,  say 
a,  and  find  the  corresponding  value  of  y,  then  give  to  x  a  new 
value,  a  +  Jx,  and  again  find  the  corresponding  value  of  j^  Sub- 
tract the  first  of  the  equations  thus  obtained  from  the  second  and 
we  have  the  value  of  Jj. 

Divide  both  sides  by  Jx  and  we  have  the  value  of  -^. 

Jx 

Then  finall}^  find  the  limit  of  this  fraction  as  Jx  approaches 

zero. 

11.  We  will  now  exemplify  the  method  in  a  few  examples. 
Firsf.  Find  D,-j^' when  j'=4.x^-\-^.  (i) 
I^et  x==a  and  represent  the  corresponding  value  of  r  by  d  and 

we  get  ^=4«'-f-5.  (2) 

Now  let  x=^a  +  Jx  and  the  corresponding  value  of  j'  will  be  the 
value  d  plus  the  amount  by  which  j/  has  been  increased,  or  b-{-Jy, 
hence  d-{-Jj'=4.(a  + JxJ'+s.  (3) 

Expanding,  b-\-Jy=/\^a^-\-SaJx-\-/^(Jx)^-\-^.  (4) 

Subtract  (2)  from  (^)  and  we  get 

Jy^^aJx^^(Jx)\      .  (5) 

Divide  ( s)  ^Y  ■^^'  ^^^  we  obtain 

-^=8a  +  4rJx;.  (6) 

Taking  the  limit  of  each  side  as  Jx  approaches  zero  we  get 

limit    (Jy\ 

or                                                D,j=8«.  (8) 

Seco7id.  Find  Y)^y  when  y=^cx--\-e.  (i) 
Let  x—a  and  represent  the  corresponding  value  ofj'by  b,  then 

b=ca^-\-e.  (2) 
Now  let  x=-a-^Jx  and  we  get 

b^Jy=.c(a^Axy-^e,  (3) 
or  expanding, 

b-^Jy^ca'-\r2acJx-\rc(Jxy-\-e.  (4) 

Subtract  {2)  from  (\)  and  we  get 

J>/=  2acJx + c(  Jx)\  (5) 


Derivatives.  171 

Divide  both  sides  of  ($)  by  Jx  and  we  get 

Jr/ 

-~-  =  2ac-i'cJx.  (6) 

Taking  the  limit  of  each  side  as  Jx  approaches  zero  and  we  get 

limit    (Jjy] 

or  D,r=2«r.  (8) 

Third.     Find  D,.^r  when  y—cx'^-\-eX'\-f,  (i) 

IvCt  x=^a  and  represent  the  corresponding  value  oi  y  by  b  and 
we  get  l^cd'-\-ea-\-f.  (2) 

Now  let  x=a-\-^x  and  we  get 

bJ^^y^c(a-VAxr^-e(a-^^x)-irf.  (3) 

or  expanding  and  arranging, 

b-^Ay=.cd'^ea^f-^(2ac^e)^X-^c(^x)\  (4) 

Subtract  (2)  from  (\)  and  we  get 

Jy^(2ac-\-e)^x^-c(^xy.  (5) 

Divide  both  sides  by  ^x  and  we  get 

■^-  =  2ac-\-e-\-cJx.  (6) 

Taking  the  limit  of  each  side  as  J.r  approaches  zero  we  get 


limit    \^y\  ,  '  .  . 

^'•O  oijl^|  =  2^^+^'  ^7) 

or  D..i'=2«r+<?.  (8) 


12.  In  what  precedes  ^^.r  has  always  been  considered  positive, 
but  Jr  may  be  negative,  in  which  case  x  is  increased  by  a  nega- 
tive quantity,  or  is  really  diminished,  so  it  may  be  more  proper  to 
call  Jjr  the  change  in  the  value  of  x  than  to  call  it  the  amount  by 
which  X  has  been  increased.  Either  way  of  speaking  is  proper 
provided  we  understand  that  the  increase  may  be  negative.  In 
any  case  Jjt  is  the  amount  that  must  be  added  to  one  value  of  ,r 
to  give  another  value  of  .r,  and  if  the  second  value  is  greater  than 
the  first  the  amount  added  will  be  negative. 

The  same  remark  applies  to  -j'. 


1/2  Algebra. 

13.  Examples.  By  the  method  already  explained  and  illus- 
trated find  the  derivative  of  the  following  expressions,  supposing 
in  each  case  that  a  is  the  value  of  x  from  which  the  increase  of  x 
is  counted  : 

/.     3.r-f2.  5.     cx^. 

2.  2>^'^-\-2X.  6.     c(x-\-\y. 

3.  (x+l)(x-{-2).         ■  7.        ~. 

4.  (x+c)\  8.      s/x. 

14.  Extension  of  Meaning  of  T).^y. 
In  Art.  10  we  found  that  when 

j=4^^  +  5,     D,.y=8« 
when  y=^cx'^-\-e,     Dj,y=2ac 

and  when  j=cx^-\-ex-\-/,     T>^y=2ac-\-e 

In  each  case  D^  r  is  of  course  a  constant  as  it  should  be  by  the 
definition  in  Art.  9,  where  T)_^y  is  defined  to  be  a  limit,  and  the 
limit  of  a  variable  is  by  definition  a  constant.  * 

In  each  case  here  noticed  D^^  is  a  constant  whose  value  de- 
pends upon  the  value  a  from  which  we  begin  to  count  the  increase 
of  X,  or,  as  we  may  say,  Ti^y  is  a  function  of  a,  while  in  Art.  5 
T>^y  was  a  constant  which  does  not  depend  upon  a. 

In  any  case  D^ji'  is  either  a  function  of  a  or  it  is  independent 
of  a,  and  when  it  is  a  function  of  a  the  a  is  the  value  from  which 
we  begin  to  count  the  increase  of  x. 

Now,  as  we  nia}^  begin  to  count  the  increase  from  a7iy  value  of 
jf,  a  is  of  course  a?ty  value  of  x\  and  so  we  may  represent  it  by 
X  instead  of  a  and  relieve  D^y  from  being  a  constant,  or  in- 
other  words,  wherever  T>^y  was  a  function  of  a  by  the  original 
definition  we  regard  it  now  and  hereafter  as  the  same  function  of 
X  that,  by  the  original  definition,  it  was  of  a. 

15.  I^et  us  now  work  out  a  case  that  was  worked  in  Art.  11, 
using  X  now  where  a  was  used  before. 

Take  ^.=  4^-4-5.-  (0 

If  we  write  x-\-Jx  in  place  of  x  and  therefore  y-j-  M'  in  place  of 
y  we  have    ■ 

j/+Jj/=4r-r+Jx/  +  5.  (2) 


Derivatives.  173 

Expanding, 

y-\-Jy=4X^+SxJx-^4(Jxr-\-5.  (3) 

Subtract  (1)  from  (t,)  and  we  obtain 

Jy=SxJx+^(JxJ'.  (4) 

Divide  both  sides  of  (4.)  by  Jx  and  we  get 

^=8-r  +  4'J-^^  (5) 

Taking  the  limit  of  each  side  as  Jjr  approaches  zero  we  get 

limit    fJy) 

or  B,y=8x.  (7) 

We  notice  that  the  result  is  exactly  the  same  as  equation  (S) 

in  the  first  example  under  Art.  10,  except  that  x  appears  here 

where  a  appeared  before. 

We  will  hereafter  proceed  as  we  have  just  done  and  will  usually 

find  D.r  y  as  a  function  of  x,  but  occasionally,  as  in  Art.  5,  D^y 

will  turn  out  to  be  a  constant. 

16.  Derivative  of  a  Constant. 

Let  j'=a  constant,  then  as  x  does  not  appear  in  the  expression 

for  r,  X  maybe  changed  at  pleasure  and  the  change  does  not  affect 

J/,  or  J-v  may  have  any  value  whatever,  but  Jy  is  always  zero. 

Jy 
Hence  -f-=o, 

^x 

therefore  D.v^r=o. 

17.  To  FIND  THE  Derivative  with  Resect  to  x  of  the 
Algebraic  Sum  of  two  Functions  of  x. 

Let  one  function  of  x  be  represented  by  u  and  the  other  by  v, 
and  let  their  sum  be  represented  by  ]';  then 

y=u-\-r.  (i) 

When  X  is  increased  by  Jx  suppose  that  u  is  increased  by  J«, 
V  is  increased  by  Jv  and  y  is  increased  by  J)',  then  after  x  is 
increased  by  J.r  we  have 

y-\-Jy=^U-[-Au-\-V-\-Av.  (2) 

Subtract  (i)  from  (2)  and  we  get 

Jr=J//-hJr.  (2^) 


174  Algebra. 

Divide  both  sides  of  (^3 j  by  Jx  and  we  get 

Jx     Jx-     Jx  *  ^^ 

Taking  the  Uniit  of  each  side  of  (4)  as  Jx  approaches  zero  we  get 
limit  j-^y\_   limit    f-^^^)    ,    limit    \  Jz>  \ 
-^x  -  olJ^rl'-^x  ^  o\-j^j-^Jx  -  oJXrj         '  ^^ 
or  D^y=D,rU-\-T>A'. 

In  the  same  way  if  y=?^  — z'  we  would  get 

The  result  may  be  expressed  thus  : 

T/ie  derivative  of  the  algebraic  sum  of  two  functions  of  x  equals 
the  algebraic  sufu  of  their  separate  derivatives. 

18.  I'o  FIND  THE  Derivative  with  Respect  to  x  of  the 
Algebraic  Sum  of  any  Number  of  Functions  of  x. 

Let  there  be  any  number  of  functions  of  x  represented  by  ?/,  v, 
7v,  etc.,  and  let  their  sum  be  represented  by  r;  then  we  have 
y=u^-v-\-iv-^  ...  rO 

Increase  x  by  the  amount  Jx  and  suppose  that  u,  v,  it\  etc., 
are  increased  by  the  amounts  J?/,  Jv,  Jw,  etc.,  respectively  andj' 
is  increased  by  Jy,  then  we  have  after  x  is  thus  increased 

yJ^Jy=U-^Ju-}-V-{--iv-{-W-j-J7t'-\-    ...  (2) 

Subtract  (i)  from  (2)  and  we  get 

Jy==J^,^J^,^J,c'-\-   .    .    .  (:,) 

Divide  both  sides  o{  (t,)  by  Jx  and  we  have 
Jy     Ju     Jv      Jw 
Jx     Jx     Jx     Jx 
Taking  the  limit  of  both  sides  of  (4)  as  Jx  approaches  zero  we 
have 

D,i'=D.,?^H-D,2'  +  D,7r4-  ...  (^) 

If  some  of  the  signs  \n  (\)  had  been  negative  the  same  process 
could  have  been  applied  and  the  result  would  have  had  negative 
signs  in  the  same  positions  as  they  appeared  in  the  original 
functions. 

The  result  may  be  expressed  thus : 

The  derivative  of  the  algebraic  sum  of  several  functions  of  x  equals 
the  algebraic  sum  of  their  separate  derivatives. 


Derivatives.  175 

19.  Examples. 

Find  the  derivative  with  respect  to  .v  of  the  following  ex- 
pressions : 

1.  2X^-\-\X^-\-X.  ^.       Jt^-f  3A--I-2. 

2.  X^ -\- X^- -\- X -\- 1 .  5.       X-^  —  X-^\. 

3.     -r''— I.  6.     x^-\-i. 

20.  'To  FIND  THE  Derivative  with  Respect  to  x  of  the 
Product  of  two  Functions  of  x. 

Let  u  and  v  be  the  two  functions  of  x,  and  let  y  be  their  prod- 
uct, then  we  have 

y—uv.  (i) 

Now  increase  x  by  J.v  and  suppose  the  corresponding  amounts 
by  which  y,  u,  and  7'  respectively  increase  are  Jr,  -^u,  and  Jz^, 
then  we  have 

y^Jy=(u-^Ju)(v+Jv).  (2) 

Expanding  (2)  we  get 

y-{-Ay=^iiv-\-uM'-\-vJu-\-JuM\  (^) 

or  y-\-Jy=uv-\-uJv-^(v-\-Jv)J/(.  ( ^j 

vSubtract  ( i )  from  (^)  and  we  get 

Jr=  uJv+  (v^  JvJJzi.  C^J 

Divide  both  sides  of  (^5  j  by  Jx  and  we  get 

Taking  the  limit  of  each  side  of  (6),  remembering  that  the  last 

term  of  the  right-hand  member  is  the  product  of  two  quantities, 

hence  its  limit  equals  the  product  of  their  separate  limits,  and 

that  J?'  approaches  zero  as  J-v  approaches  zero,  hence  the  limit  of 

v-\-J2'=7\  we  get 

limit    f  JjM  ,.     .    Jz'  ,      ,.     .    J// 

J-v  ^  o1j^|  =  ''  l^"^^t  3^  +  z^  hmit  --,  (y) 

or  D,y=2iD,z'i-vDji.  (S) 

21.    To  FIND  THE   Derivative  of  the   Product  of  any 
Number  of  Functions  of  x. 

First,  take  three  functions  of  -v,  say  «,  z\  and  zc,  and  let  y  be 
their  product,  then  we  have 

y=uz'Zji'.  (i) 


176  Algebra. 

Let  vw=v',  ih.^n y=uv',  and  hence 

D,.jj/=  v'B^u  +  iijy^v'.  (2) 

But  Tij)'=zvT>,v-\-vT>.,zv,  (^) 

hence  by  substitution  in  (^2  j  we  have 

Now  if  we  had  any  number  of  functions  of  x,  say  u,  r,  w,  .  .  . 
and  if  we  let  j'  be  their  product  we  have 

y=iwivz  ...  (\) 

Let  the  product  of  all  the  functions  after  the  first  be  represented 
by  a  single  letter,  that  is,  let 

v'=^vwz  .    .    . 
then  y=uv'. 

Find  D.,.y  as  the  product  of  two  functions.     Then 

T>,y=v'T)ji  +  uT>,,v'.  (2) 

Find  D^t''  by  letting  v'=vw\  where  w'  represents  the  product 
wz  .    .    . 

Substitute  the  value  thus  found  in  (2).  The  result  will  con- 
tain one  temi  involving  T).,w'. 

Find  the  derivative  by  considering  iv'  to  be  the  product  of  hvo 
factors. 

Continue  this  process  until  finally  we  reach  the  product  of  the 
last  two  factors  of  the  expression  with  which  we  started. 

The  result  may  be  stated  thus  : 

The  derivative  with  respect  to  x  of  the  product  of  a7iy  yiiunber  of 
functions  is  equal  to  the  sum  of  all  the  products  obtained  by  multi- 
plying the  derivative  of  each  factor  by  the  product  of  all  the  other 
factors. 

If  the  equation  here  described  be  divided  through  by  the  prod- 
uct of  all  the  given  functions,  the  result  may  be  represented  in 
quite  a  convenient  form,  viz  : 

■"  -  -h— ^4--  — -h— ^+  ... 


y  21         V  w 

22.  ExAMPLKs. 

Find  the  derivative  with  respect  to  x  of  the  following  expres- 
sions without  performing  the  multiplications  indicated  : 
7.     (x-\-i)(x-\-2)  compare  with  example  4,  Art.  19. 


Derivatives.  177 

2.  (Af—x-j-i)(x^i)  compare  with  example  6,  Art  19. 
J.  (x^-\-x-{-i)(x—iJ  compare  with  example  3,  Art,  19. 
4.     (x'-j-j)(x^-j-ax-\-d), 

23.  ^o  FIND  THE  Derivative  with  Respect  to  x  of  the 
Quotient  of  two  Functions  of  x.. 

Let  u  and  2'  be  the  given  functions  of  x,  and  let  y  be  their  quo- 
tient, then  we  have 

V 

From  (i),  by  multiplying  by  v,  we  get 

u=zy,  (2) 

hence  T),.u=i>D_,.j'+y'D^v,  (;^) 

u 
or  Tiji=vY)^y-\-~'V^^v.  (^) 

V 

Multiply  both  sides  by  v  and  we  get 

vJ^^u^v'^jy^y+ul^.v.  (s) 

Transposing  and  dividing  by  z'-"  we  get 

Expressed  in  words  this  is 

T/ze  derivative  of  a  fraction  equals  the  denominator  into  the  deriv- 
ative of  the  numerator  minus  the  numerator  into  the  derivative  of 
the  deyiominator  all  divided  by  the  square  of  the  denominator. 

24.  Examples. 

Find  the  derivative  with  respect  to  x  of  the  following  ex- 
pressions : 

x^-\- 1 

1.  '  compare  with  example  5,  Art.  19. 

X-Y  I 

'  x^—Sx^'+iix—S  .,  ,  * 

2.     — - — compare  with  example  4,  Art.  ig. 

■^     3 

x+j  • 

^'       X'+l' 
A— 22 


178  Alge^bra. 

25.  To  FIND  THE  Derivative  with  Respect  to  x  of  a 
Function  of  Another  Function  of  x. 

Suppose  y  is  some  function  of  3,  and  .z  is  some  function  of  .r, 
then  ultimately  y  is  a  function  of  x,  hence  it  has  a  derivative 
with  respect  to  x. 

But  as  y  is  directly  a  function  of  z  it  has  a  derivative  with  re- 
spect to  z. 

Moreover,  as  s  is  a  function  of  x  it  has  a  derivative  with  respect 
to  -v. 

We  hav^e  identically 

Jx      Jz     Jx  '     ^ 

Taking  the  limit  of  each  side  as  Jx  approaches  zero,  remem- 
bering that  the  limit  of  the  product  of  two  variables  equals  the 
product  of  their  limits,  we  have 

limit    (  dy  \  _   limit    \  Jy  ]        limit    \  -Jf  / 
^-r  ;  o  ]  jjc  \  ~'^-^^  ^  o)  j^  \    '  Jx^    o  i  jjt^  (    '  ^^ 

Now  z  being  a  function  of  x  we  may  write 

2=f(x). 
and  if  x  be  increased  by  Jx  we  have 

z-^Az^f(x-{-Jx), 
and  from  this  it  is  evident  that,  as  Jx  approaches  zero,  Jz  must 
also  approach  zero.     Hence 


t    |Jj.)         limit    |Jj.| 


Substitute  from  (t^)  in  (2)  and  we  have 

limit    \  Jy\  _   limit    f  Jy  ]         limit    \  Jz  } 

The  left-hand  member  of  (/\)  is  D,j';  the  first  factor  of  the 
right-hand  member  is  T>~y,  for  it  is  just  the  same  as  the  left-hand 
member  except  that  z  everywhere  takes  the  place  of  x;  and  the 
second  factor  of  the  right-hand  member  is  D,  2-. 

Hence  T>^y=T>,y  .Ti^z.  (^) 

If  y=pf  and  .5'=jr'-f-2,  then 

l)^y=2z  and  D,z=2x. 
Hence  by  equation  (s) 

D ,  y=2z  .  2X=  ^zx=  ^x(x^  -\-2)  =  /\.x '  +  8x. 


Derivativks.  179 

It  is  easy  to  see  that  this  result  is  correct,  for  in  the  equation 
1'=^-,  substitute  the  value  of  r  and  we  have 

Hence  D,  r=4.r^+8-r. 

26,  Examples. 

Find   the  derivative   with   respect  to  x  of  the   following  ex- 
pressions : 

/.      (x^  +  ax-hdj\ 

5.  ^x=  +  3A--f2/-5rA-+3Ji-H-2;4-5. 

6.  2(x'-Y)^-\-^(x'-ir-^(x^-l). 

27.  To  KIND  THE  Derivative  with  Respect  to  x  of  any 
Positive  Integral  Power  of  x. 

Let  y=-x".  '  (ij 

Give  to  X  the  value  x-\-Jx  and  we  get 

r-hJj/^/jr-f  J.rj".  (2) 

Expanding  the  right-hand  member  of  (^2  j  we  gel 

r+jv=.v'--f;/A-"-'jA-+ -''"''" 'V-Y-^t-/+  .  .  .  +r-'-r;-.  (?>) 

Subtract  (i)  from  ^3^  and  we  get 

Divide  both  sides  of  ( /^)  by  -  .r  and  we  get 

Taking  the  limit  of  each  side  as  Jx  approaches  zero  we  have 

D^y=nx"-\  (6) 

Reasoning  exactly  as  above  we  could  show  that  when   ]'=<?.v", 

Y>,y=nax"~\ 
This  fonnula  may  be  expressed  in  words  thus — 
The  deriratiTC  with  respect  to  x  of  ax"  is  found  by  multiplying 
the  exponent  by  the  eoefficierJ  and  reduci^ig  the  exponeiit  \ . 


i8o  Algebra. 

It  is  to  be  noticed  that  this  formula  applies  to  the  derivative 
with  respect  to  x  of  a  power  of  x.  Of  course  any  other  letter  be- 
sides X  could  be  used  to  denote  a  variable. 

Thus,  whenj/=«r",  T>^y=na2"~\ 

But  we  must  be  careful  not  to  use  this  formula  to  find  the  de- 
rivative with  respect  to  some  quantity,  of  a  power  of  some  other 
quantit3%  Or,  in  oth,er  words,  in  order  to  be  able  to  use  this  form- 
ula the  quantity  which  is  raised  to  a  power  must  be  same  as  that 
with  respect  to  which  the  derivative  is  taken. 

28.  "To  FIND  THE  Derivative  with  Respect  to  x  of  any 
Negative  Integral  Power  of  x. 

Let  j,=a-»=l  (i) 

T).y=^^^^^^^^  by  Art.  23.  (2) 

Simplifying,  remembering  that  the   derivative  of  a  constant  is 
zero,  we  get 

^^y=--^.n=-^^'    '  (3) 

It  may  be  objected  to  this  method  that  we  have  used  the  form- 
ula for  the  derivative  of  a  fraction  whose  numerator  is  i  when 
that  formula  supposed  that  numerator  and  denominator  were 
each  functions  of  x. 

We  may  then  take  j=   "„^^ 

and  uo7t'  use  the  formula  of  Art.  23  and  we  get  as  before 

D,  r= — nx~"~\ 

It  easily  follows  that  if 

y=:ax'~", 
then  D,^y=—nax~"~\ 

Here,  as  in  Art.  27,  in  order  to  use  the  formula  the  quantity 
raised  to  a  power  must  be  the  same  as  the  one  with  respect  to 
which  the  derivative  is  taken. 

We  may  express  this  formula  in  words  thus — 

T/ie  derivative  ivith  respect  to  x  0/  ax~"  is  found  by  ?nultiplyin^ 
the  expojient  by  the  coejficie^it  and  reducing  the  exp07ient  i. 


Derivatives.  i8i 

29.    I'o  FIND  THE  Derivative  with  Respect  to  a-  of  a 
Fractional  Power  of  x. 


ut 

z=xf 

(I) 

and  let 

j/==^^=A-^ 

(2) 

Then 

T>,y-=T>,y.D,z,  Art.  25. 

(3) 

But 

J}^y^qz'^-\ 

(A) 

hence 

I)^y=gz''-\  B^z. 

(5) 

But  from  (2) 

D,j'=/>a:^-', 

(6) 

hence  from  (^) 

'  and 

(^) 

qz"-'.  J}_,z=px^-\ 

(7) 

Divide  by  qz', 

which  is 

the  same  as  qx^,  and  we  get 

z-'D,z=^x-\ 

(^) 

p 
Multiply  the  left  membei   by  z  and  the   right  member  by  a"'', 

which  is  the  equal  of  z,  and  we  obtain 

p  -t-, 
T>,z—~x'f 

The  same  reasoning  would  show  that  if  r=«A:^,  then 

_         ap   ^-i 
T>^y=^  x'' 
9 
Hence,  as  in  the  two  preceding  articles,  the  result  is  obtained 
by  multiplying-  the  exponent  by  the  coefficient  a?id  reducing  the  ex- 
po7ient  I . 

30.  Examples. 

Find  the  derivative  with  respect  to  x  of  the  following  expres- 
sions : 

I.     ( x-—x-\- 1 ) -\- 2( x^—x-\- 1). 

ix^—i]   ,     (x^—i 
Vx^—\ 


5.     {x-\-sJ  \-x^)\ 


l82  A1.GEBKA. 

(). 
7. 


9- 


JO. 


\/  i+.r—  \/  i—x 


1 1  +  V I  —'^" 

I      i        Jill         1 


\/  a-\-A 


s/  a-\-  sf  X 


J;iy 


\        .r     X' 

12.     (a-j-xj".  (d+x)'-. 


CHAPTER  XIV. 

SERIES. 

1.  A  definition  of  a  series  was  given  in  XII,  Art.  4,  and  it  was 
there  noticed  that  infinite  series  are  divided  into  the  two  classes 
of  convergent  and  divergent.  Convergent  series  have  definite 
limits  as  the  number  of  terms  is  increased  without  limit,  but  from 
their  nature  divergent  series  are  wholly  indefinite,  and  hence  //  is 
not  safe  to  use  divergent  series  or  to  base  any  reasoning  upon  them. 

In  all  that  follows,  and  indeed  in  all  that  precedes,  it  is  to  be 
understood  wherever  infinite  series  are  used  that  the  results  hold 
as  long  as  all  the  series  used  or  obtained  are  convergent. 

In  many  cases  a  series  is  convergent  or  divergent  according  to 
the  value  of  some  letter  in  the  series,  and  it  is  always  understood 
in  such  cases  that  the  letter  concerned  is  limited  to  those  values 
which  make  the  series  convergent,  and  no  inference  is  to  be 
drawm  for  any  other  value. 

It  would  be  fortunate  if  some  simple  and  universal  criterion 
were  known  whereb}-  we  might  determine  w^hether  any  given 
series  is  convergent  or  divergent,  but  unfortunately  no  such  cri- 
terion has  been  found.  There  are,  however,  some  cases  in  which 
we  can  determine  whether  a  series  is  convergent  or  divergent  and 
we  gi\'e  a  few  of  these. 

2.  L-t  the  terms  of  a  series  be  represented  by  u^,  u^,  u,,  etc., 
in  each  case  the  subscript  being  the  same  as  the  number  of  the 
term;  and  let  R^  be  the  remainder  after  the  first  term,  R„  the  re- 
mainder after  the  second  term,  R,  the  remainder  after  the  third 
term,  etc. ;  in  each  case  the  remainder  after  any  number  of  terms 
are  taken  is  represented  by  R  with  a  subscript  equal  to  the  num- 
ber of  terms  already  taken;  and  further  let  the  sum  of  any  number 
of  terms  be  represented  by  S  with  a  subscript  equal  to  the  num- 
ber of  terms  taken,  /.  e.  the  sum  of  two  terms  will  be  represented 
bv  S.,  the  sum  of  three  terms  bv  S,,  and  so  on. 


184  Algebra. 

3.  With  the  notation  just  explained,  the  sum  of  a  series  which 
has  a  limited  number  of  terms  will  be  represented  by  S,^  +  R^, 
whether  ^  is  i  or  2  or  3  or  any  other  number  not  exceeding  the 
whole  number  of  terms  of  the  series. 

In  an  infinite  convergent  series  S„  approaches  a  limit  as  n  in- 
creases without  limit,  and  the  value  of  this  limit  is 

S^  +  R,^, 
where  q  is  any  positive  whole  number  whatever.     It  is  easy  to 
see  in  this  case  that  ^..^  o  as  n  increases  without  limit. 

In  an  infinite  divergent  series  S„  does  not  approach  any  limit 
neither  does  R,,  approach  any  limit,  and  S^  +  R;^  has  no  definite 
value  at  all. 

4.  It  is  evident  that  a  series  cannot  be  convergent  unless  after 
a  certain  number  of  terms  are  taken  the  successive  terms  grow 
smaller  and  smaller,  or,  in  other  words,  unless  ?/„;:  o  as  n  in- 
creases without  limit.  But  while  this  is  necessary  it  is  not  suf- 
cient,  for  a  series  viay  be  divergent  and  still  u,,'^  o  as  n  increases 
without  limit. 

Take  for  example  the  series 

i  +  R-Hi+  .  .  .  ■ 

where  the  nth  term  is  — ,  which  evidently  approaches  zero  as  n 

increases  without  limit. 

If  this  series  be  grouped  thus  : 

+  a  +  TV  +  TV  +  TV  +  A  +  T'4+TV+TV)+    •     ■     • 

then  in  no  group  is  the  sum  less  than  ^  and  as  there  are  an  un- 
limited number  of  groups,  the  series  evidently  does  not  approach 
any  limit,  but  increases  without  limit  as  the  number  of  terms  in- 
creases without  limit,  therefore  the  series  is  divergent. 

5.  ThkorKm.  a  series,  all  of  whose  ter?ns  are  positive,  is  diver- 
gent if  nu,^  does  not  approach  zero  as  n  increase's  without  limit. 

Since  all  the  terms  are  positive,  7iii,^  is  positive,  and  since  7111,^ 

does  not  approach  zero,  we  may  take  r  a  quantity  so  near  zero 

r 

that  n2i,^r,  then  ?/„>—. 

n 


Series.  185 


bimiiarilv,  /^,.i>     ,     ,     ?^;,^9>      ,     ,  etc. 
«-f  I  "■     n-\-2 


Hence 


;2      ?/  4- 1      n-\-2 


But  the  quantities  in  the  parenthesis  form  the  terms  of  the 
series  i-{-\  +  \-\-  .  .  .  after  the  ?ii\i  term,  and  this  latter  series 
has  been  shown  to  be  divergent,  or  in  other  words,  the  quantity 
in  the  parenthesis  increases  without  limit, ^  and  therefore 

increases  without  limit ;  therefore  the  series  is  divergent. 

6.  Theorem  .  If  the  terms  of  a  series  are  alternately  positive  and 
negative  and  after  a  certain  number  of  terms  each  term  is  numeri- 
cally less  than  the  preceding  one^  and  the  n  th  term  approaches  zero, 
as  n  increases  ivithout  limit,  the  series  is  convergent. 

Let  the  series  be 

u^-—u^^-{-u^  —  u^-]r  .    .    . 
and  let  the  series  be  represented  by  S ;  then  we  may  write  either 

S  =  S,-f  r«,,+  i— ?^^+.J+r2^./+3— 2Vf4^+    ...  (t) 

or       s=Sv,i— r^Vf'i— ?^^+:j— r^^^+4— ?wJ—  .  •  .      (2) 

After  a  certain  number  of  terms,  say  k,  each  term  is  less  than 
the  preceding  one,  so  if  q  be  larger  than  k,  each  parenthesis  in 
(i)  and  also  in  (2)  is  positive,  and  therefore 
from  {1)  S>Sy, 

and  from  (2)  S<S^+i. 

Thus  we  see  that  S  is  intemie'diate  in  value  between  the  two 
definite    quantities    S^   and    S^^i,  which    two    quantities    diifer 

by  /Vti- 

Similarly,  whatever  positive  whole  number  be  represented  by 
r,  we  get 


S>S 


and  S<Sy.(2'-M* 

Now  S^.,2.>S,/ 

and.  ^4't  2''+ 1  "^^^H  1  • 

Therefore  S  is  intermediate  in  value  between  two  quantities, 
the  larger  of  which  grows  smaller  and  the  smaller  of  which  grows 
larger. 


186  Algebra. 

Moreover,  S,,,  2-i  i  and  S^,  2 r differ  by  /^„4.2hi>  which  approaches 
zero  as  r  increases  ;  therefore  the  two  quantities  between  which  S 
is  always  found  approach  equality  as  r  increases.  Therefore  S 
has  a  definite  value,  or,  in  other  words,  the  series  is  convergent. 

7.  Theorem.  If  all  the  terms  of  a  series  are  positive  and  after  a 
certain  7inmber  of  terms  each  term  is  less  than  the  one  before  it  and 
the  limit  of  the  nth  tert?i  is  zero ;  the?i  if  the  limit  of  the  ratio  of  the 
^-fi^i^f/i  tern  I  to  the  nth  term  is  less  than  i  the  series  is  convergent. 

If  all  the  terms  are  positive  and  after  a  certain  number  of  terms 
each  term  is  less  than  the  preceding  one,  then,  anywhere  after 
this  certain  number  of  terms,  the  ratio  of  any  term  to  the  preced- 
ing one  is  positive  and  less  than  i.  Now,  since  each  of  these 
ratios  is  less  than  i  and  the  limit  of  the  ratio  is  less  than  i,  we 
may  take  some  quantity,  /',  less  than  i  but  so  near  i  that  each 
ratio  will  be  less  than  k. 

Hence  '      <a'     -•.      u„^^<^ku„ 

u„ 


«+2 


//, 


■Ck     . ' .      n„^_ 3  <iku.,  f  2  <ih'^  u,^ 
etc. 


Therefore 

^^Hi+'^':-2+?^"+3+    •    •     .   <ufk^k"-+k'^+   .    .    .  ) 
or,  adding  h„  to  each  side  of  the  inequality, 

/^,  +  /V:  1  H-«^.,  2+^^;H3+     •      •      •     <U  f  I -^  k -\- k'^ -\- k'^ -{-    .      .      .    ) 

But  when  /w'<i 

i-j_/:4-/>2^/.:i^  _        ^  ^^  See  XII,  Art.  8. 

I— A- 

and  this  is  a  definite  quantity. 

Therefore  the  right-hand  side  of  the  last  inequality  is  a  definite 
multiple  of  u,„  and  u„  ^  o  ;  therefore  the  right-hand  side  of  the 
last  inequality  approaches  zero  ;  and  as  the  left  side  is  less  than 
the  right  side  and -neither  side  can  be  negative,  therefore  the  left- 


Series.  187 

hand  side  of  the  last  inequality  approaches  zero  ;  therefore  the 
remainder  after  ri—  i  terms  approaches  zero  :  therefore  the  series 
is  convergent. 

8.  In  the  theorems  of  the  two  preceding  articles  the  student 
should  note  the  force   of  the  words  ''after  a  certain  number  of 
terms. ' '     The  first  few  terms  of  a  series  may  not  give  any  indica- 
tion as  to  whether  the  series  is  convergent  or  divergent.     Take, 
for  example,  the  series 

where  the  rth  term  is  rx''~\  and  suppose  x=-^-^ ;  then  the  succes- 

sive  terms  grow  larger  up  to  the  ninth  term,  which  =^3.     The 

tenth  term  has  the  same  value  as  the  ninth,  but  every  term  after 
the  tenth  is  less  than  the  preceding  one.  Moreover,  as  n  increases 
without  limit,  the  //  th  term  approaches  zero  and  the  ratio  of  the 


(7i-\- 1  jth  term  to  the  7^th  term  equals    i  +       x,  and  this  evidently 

L        ^^  J 
approaches  x  as  a  limit.     Hence  the  series  is  convergent. 

9.  Theorem.  A  series  is  convergeyit  if  the  series  obtained  by 
making  all  its  terms  positive  is  convergent. 

Let  the  limit  of  the  sum  of  the  positive  terms  be  represented  by 
Uj,  and  the  limit  of  the  sum  of  the  negative  terms  be  represented 
by  U,;  then  the  limit  of  the  sum  of  the  series  will  be  U,— U^. 

Now  consider  a  new  series  formed  from  the  given  series  by 
making  all  its  terms  positive  ;  then  the  limit  of  the  sum  of  this 
new  series  will  be  U,  +  U^,  and  as  this  new  .series  is  convergent  by 
hypothesis,  U,-f  U,,  has  a  definite  value.  Again,  as  U,  and  U^  are 
both  positive  and  as  their  sum  has  a  definite  value,  therefore  each 
of  these  quantities  U^  and  U^  has  a  definite  value,  therefore  their 
difference,  U^— U^,  has  a  definite  value  ;  therefore  the  .series  is 
convergent. 

10.  Theorem.    The  series 

a^-\-a^x-\-a^x^-\-a^x'^-\-   .    .    . 
is  convergent  when  a'-<i,  unless  a„  increases  without  limit  as  n  in- 
creases without  limit. 


1 88  Algebra. 

We  may  consider  all  the  terms  positive,  for  if  some  were  nega- 
tive we  could  form  a  new  series  all  of  whose  terms  were  positive 
and  conduct  the  reasoning  upon  the  new  series,  and  if  this  new 
series  were  convergent  the  original  series  would  be  so  by  Art.  9. 

Since  we  may  consider  all  the  coefficients  positive  and  since  a„ 
does  not  increase  without  limit  we  may  take  b,  a  quantity  greater 
than  the  greatest  of  the  coefficients,  then 

a^-\-a^x-^ax^-\-   .    .    .   <^b-\-bx-\-bx^-\-   ... 

But  the  right-hand  side  of  this  inequality  equals ,(see  XII 

I — X 

Art.    8,)   /.  e.  the  right-hand  side  has  a  definite  value,  therefore 

the  left-hand  side  also  has  a  definite  value  ;  hence  the  series  i-s 

convergent. 

II.  TheorKm.  If  we  have  given  a  series  such  that  after  a  certain 
number  of  terms  each  term  is  less  than  the  correspondi7ig  term  0/ 
some  se7  ies  which  is  known  to  be  convergent,  then  the  give^i  series  is 
convergent. 

Let  the  given  series  be 

u^-\-u^-\-uA-u^-^  ...  (i) 

and  let  the  series  known  to  be  convergent  be 

t\+^^,+t'3  4-t\+  ...  (2) 

and  suppose  each  term  after  the  rth  in  (i)  to  be  less  than  the  cor- 
responding term  in  (2). 
Since  (2)  is  convergent, 

^V+i+^r+ 2+^^  +  3  +  ^^1-4+     •      •      • 

approaches  a  definite  limit  as  the  number  of  terms  increases  with- 
out limit,  and  since  ?/,.j  ^  <;c'..+  i,  ?/;  + 0  ^^V-i  2 >  2^;  +  3>'^r-f  3,  etc., 
therefore  Ur.^  ^  -\-  u^j^  2  +  ^'-+  3  +  ^'-+  4  +  •    •    • 

approaches  a  definite  limit  as  the  number  of  terms  increases  with- 
out limit.  Now  as  the  sum  of  the  first  r  terms  of  (i)  is  a  definite 
quantity  and  the  sum  of  the  terms  after  the  rth  approaches  a  def- 
inite limit,  it  follows  that  the  whole  series 

U  -{-11  -\-2l  ^u  -\-   .     .     . 
approaches  a  definite  limit  as  the  number  of  terms  increases  with- 
out limit,  or  in  other  words,  the  series  is  convergent. 


SERIEvS.  189 

12,    Examples.     Determine   whether    the    following   infinite 
series  are  convergent  or  divergent  : 

/.      1 4- '+'..+  ',+  .    .    . 
2      3'     4^ 

I"        2^         ^^ 

i^        !3        1 4 
X      X'       x^ 

3'     1 4-"- 4-- -4-"  ,4-  .    .    .  when  .i-<i. 
23-      4^ 

/.      i4-'7—  4-  , V  ,  -   4-  .    .    .  when.v<i. 

1 2        !3        ^4 

5".        —      .     4-      ,       —      ,       4-  .    .    .    .  when  .v  and    a    are 
.1-     x-\-a     x-\-2a     -1+3^ 

positive. 

6.       ' 1-4-         -f         4-  .    .    .  when.r<i. 

1.2     2.3     3.4     4.5 

-        I  _. I 4. I I       _. 

^-     xy     (x-{-i)(y-i-iy(x-\-2Xy-^2)     (x-hsJO'+sJ         '  ' 
Avhen  .V  and  r  are  positive  quantities. 

X      .     X-         x"^    ,     A~* 

<S\  4-         H .4 ^4-.    .    .whenA-<i. 

1.2     3.4     5.6     7.8 

2"  '^-  4."* 

5/.      1+         +  ,      +T-+  •    •    • 


U 


.  -  J^ J3- J^ Ji 
-  »M-  •  •  ■ 

I         2""       '^' 

2-     3^     4'' 

.V"  X"  .V* 

/?.     Show  that  i4--v4-     -4-  ,      4--, — h  .    .    .is   convergent 
\1        1,3        U 
for  all  values  of  .v. 

2'        3'        4" 
/^.     Show  that  1 4-        4-        H h  .    .    .is   convergent    or 

234 

all  values  of  .v. 


I90  Algebra. 

Taylor's  formula. 

13.  Taylor's  formula  is  a  very  general  one  that  enables  us  to 
obtain  the  development  of  a  function  of  a  binomial  .^-f /?  arranged 
according  to  positive  increasing  powers  of  //.  Whether  the  func- 
tion be  integral  or  fractional,  rational  or  irrational,  it  matters  not ; 
indeed  a/ij'  function  of  a  binomial  x-\-/i  which  is  capable  of  being 
expressed  in  the  form  of  a  series  arranged  according  to  positive 
increasing  powers  of  //  can  be  thus  expressed  by  means  of  Ta3'lor'8. 
formula. 

Sometimes  the  series  will  be  finite  and  sometimes  (indeed 
usually)  infinite,  but  in  case  the  series  is  infinite  it  must  be  re- 
membered that  the  series  and  the  function  cannot  be  considered 
equivalent  unless  the  series  is  convergent. 

Before  we  can  take  up  TaA'lor's  formula  it  is  necessary  to  ex- 
plain what  is  meant  by  successive  derivatives  and  to  give  a  theorem 
not  given  in  the  chapter  on  derivatives.     These  we  now  take  up. 

14.  SuccKSSivE  Derivatives.  If  we  represent  a  function  of  x 
hy  f(x)  w^e  may  find  the  derivative  with  respect  to  x  of  this  func- 
tion of  X,  and,  as  the  result  is  usually  another  function  of  x,  .we 
may  represent  it  by  f'(x).  Again,  we  may  find  the  derivative 
with  respect  to  x  oi  f'(x)  and  may  represent  this  by  f!'(x). 

Thus  we  see  that  f"(x)  is  the  derivative  wath  respect  to  .r  of 
the  derivative  with  respect  to  ..v  o{f(x).  This  is  called  the  second 
derivative,  with  respect  to  x  of  f(x),  and  is  represented  by  the  nota- 
tion T)lf(x). 

We  may  find  the  derivative  with  respect  to  .v  oi/"(x)  and  rep- 
resent the  result  by  f"(x).  Thus  we  say  that  f'"(x)  is  the 
derivative  with  respect  to  x  of  the  second  derivative  with  respect 
to  X  oi  f(x).  This  is  called  the  third  derivative  with  respect  to  x 
oi  f(x),  and  is  represented  by  the  notation  Vi\f(x),  and  so^pn. 

For  example,  if  we  take  ax"  as  the  function  of  .v  we  start  with » 
then 

Ti^ax"=nax"~'^ 

T>^.ax" =T>^nax"~'=n(  n—  I  )ax"~^ 
D3<7.r"=  D^n(n  —  i  )ax"~''=  n(n  —\)(n  —  2)ax"~^ 
etc. 


Series.  191 

15.  Examples. 

/.     Find  5  successive  derivatives  of  x\ 

2.     Find  4  successive  derivatives  of  x^-\-x*'-\-x^-\-y-\-x-\- 1. 

J.     Find  3  successive  derivatives  of 

^.     Find  3  successive  derivatives  of  \^  i-j-x. 

5.  Find  3  successive  derivatives  of  V  i  +  2x. 

6.  Find  3  successive  derivatives  of  sf  \-\-x". 


Find  3  successive  derivatives  of 


—  x 


16,  Theorem.  In  a  fiindiou  of  a  binomial  x-\-/i,  sayf(x-\-h), 
ihe  derivative  with  respect  to  x,  wheri  h  is  regarded  consta7it,  is  equal 
to  the  derivative  with  respect  to  h  ivhen  x  is  regarded  consta?it. 

l^etv=f(x-{-hJ,  and  let  x-\-h=2,  then 

D,  r=D,  r.  D,5.  See  XIII,  Art.  25,  Kq.fs) 
Eut  D,,2'=i. 

Hence  Dj'=D..j'.  (i) 

Again,  D/,j=D,j'.  I)i,z.  SeeXIII,  Art.  25,  Eq.(5) 

But  T>,,z=i. 

Hence  Df,v='D,v.  (2) 

From  (i)  and  (2)  it  follows  that 

17.  Taylor's  Formula.  We  are  now  prepared  to  take  up 
Taylor's  formula. 

If  f(x  +  h J  can  be  developed  into  a  series  arranged  according  to 
positive  increasing  powers  of  //,  let  us  assume 

/('.v-f/^;=A,-f A//  +  Ay/--fA>^+  .    .    .  (i) 

A\iiere  A  ,  A,,  A..,  etc.,  do  not  contain  /;,  but  are  in  general  func- 
tions of  .r. 

Take  the  derivative  with  respect  to  x  of  each  side  of  ( i )  and  we 
have 

D,J(x-\-h)==D,,Aj-hB,,A^-^lrD,A^  +  h'D,A^+  .    .    .     (2) 

Also  take  the  derivative  with  respect  to  //  of  each  side  of  (i) 
and  we  have 

D„/rr+//;  =  A,  +  2A//-f 3A/r  +  4A/^+  .    .    .  (3) 


192  Algebra. 

By  Art.    16  these  two   expressions   must   be   equal,   therefore 
equating  coefficients  of  like  powers  in  (2)  and  (3)  we  get 


A=DA 

(4) 

2A^=D,A,     . 

■.     A  =,VD,A  =-D?A„ 

(5) 

3A=D,A,     . 

•.     A=iD,A=_'_D.?A. 

■    0 

(6) 

4A=D.,A,     . 

•.     A  =iD.A.=  ^^D;^A 

^     '    ■     ^     2.3.4 

(1) 

5A=DA     • 

•.     A.=iD,A=       ^      D^A 
^              '     2.3,4.5    ■      ' 
etc. 

(8) 

NowMf  we  make  h  =  o  in  (i)  it  is  easy  to  see  that  X^=f(x), 
w\\.&r^f(x)  means  the  same  function  of  ,r  that  the  given  function 
is  of  .r+/?,  or  in  other  words,  f(x)  is  what  the  given  function  be- 
comes when  h  is  put  equal  to  zero. 

If,  in  (i),  we  substitute  /(^xj  for  A.,  and  for  the  coefficients  of 
the  various  powers  of//  the  values  found  in  equations  C4)  to  (8), 
we  get 

f(x^-h)==f(x)  +  hJ}jX^x)  +  ~^''^I)lf(x)-^    ^'lj)-^f(x)-^ 

This  result  is  Taylor's  formula  and  is  often  spoken  of  as  Ta}'- 
lor's  theorem. 

18.  AppIvICATion  of  Taylor's  Formula. 

Let  us  develope  (x-\-hf  by  Taylor's  formula. 

Here  f(x-\-h)  =  (x-\-h)\ 

Therefore  f(x)=x\ 

Finding  the  successive  derivatives  of  x^  w^e  get 

TiJ(x)  =  6x\  T}lf(x)  =  6  .  5Jr^ 

mf(x)==6.SA^\  Dyr-^;  =  6.5.4.3-x-=, 

Dj/f.^;=6.5.4.3.2Jt^         D«/r-^;  =  6.5.4.3.2.i, 

and  every  derivative  after  the  seventh  will  equal  zero.     Therefore 
by  substitution  in  Taylor's  formula  we  get 

(x-\-h)'=x'+6xyi  +  ^ '  ^x'Jr  +  ^'~^'^x%^ 

+  6,5.4.3^.^,^6,^4,3:2^^,^6.5^,2^,^ 
2.3.4  2.3.4.5  2.3.4.5.6 

or       (x^Iif=x^-\-6x^h-^iSx'h'-\-20xyp-{-iSxVi'  +  6xh''-^x\ 


Series.  193 

This  result  is  seen  to  be  the  same  as  that  obtained  by  a  direct 
application  of  the  Binomial  formula,  which  of  course  is  as  it  ought 
to  be. 

As  a  second  example,  let  us  develope  ^ x-\-/i  by  Taylor's 
formula. 

Here  f(x+h)=:(x-^h)^ . 

1 
Therefore  f(x)=x'^. 

Finding  the  successive  derivatives  of  x^  we  get 

etc. 
Making  the  substitutions  in  Taylor's  formula  we  get 

j_  _x  _ii  _A  —L 

s/x-^h=x--\-\x   '^h—\x  Vz^+yV-^  V/^— yfg.r  V/'-h   .    .    . 

If  in  this  equation  we  make  jr=  i  we  get 

If  in  this  equation  we  change  the  sign  of  h  we  get 

Compare  this  with  the  result  obtained -in  XII,  Art.  12. 

It  was  stated  in  Art.  13  that  Taylor's  formula  could  be  used  to 
develope  a7iy  function  of  a  binomial  which  is  capable  of  being  de- 
veloped into  a  series  arranged  according  to  positive  increasing 
powers  of  one  of  the  quantities.  It  is  indeed  a  matter  of  substi- 
tution, but  care  must  be  taken  that  the  substitution  be  such  that 
the  development  obtained  is  arranged  according  to  positive  in- 
creasing powers  of  the  proper  quantity. 

If,  for  example,  we  wish  to  develope  V^'+i  into  a  series  ar- 
ranged according  to  positive  increasing  powers  of  .r,  it  might  at 
first  appear  that,  in  the  development  of  sf  x-\-h,  we  could  simph' 
make  h=i  ;  but  this  would  give  us  a  series  arranged  according 
to  positive  increasing  powers  of  i,  not  x.  The  proper  course  is  as 
follows  :  First,  develope  V  x-\-h  according  to  positive  increasing 
powers  of  h  ;  then  in  this  result  make  .i  =  i  and  we  have  the  de- 
velopment oi  \/i-\-h  arranged  according  to  positiv^e  increasing 
powers  of//;  then,  finally,  in  this  result,  change  //  into  .i*  and  we 
obtain  the  result  sought. 


V 


194  Algebra. 

19.  Binomial  Theorem  for  any  Exponent. 

Let  us  apply  Taylor's  formula  to  the  development  of  (x-\-h)" 
according  to  positive  increasing  powers  of  //,  where  n  is  either 
positive  or  negative,  iyitegral  or  fractional. 

Here,  then,  f(x^h)  =  (x^h)\ 

Therefore  f(x)=^x". 

Finding  the  successive  derivatives  of  x"  we  obtain 
jyJ(x)  =  nx"-\  T)lf(x)  =  7i(?i-i)x"-% 

J)lf(x)  =  n(?i-i)(n-2)x"-\  B^/(x)  =  n(n-i)(n-2)(n-3)x"-' 

etc. 
Therefore  by  substituting  in  Taylor's  formrfla  we  get 

_^^^,Xn-22(n-3)^„_^^^,^  ...     (I) 

li 

Thus  we  arrive  at  the  Binomial  formula,  where,  however,  the 
exponent  is  not  restricted,  as  in  Chapter  X^,  to  being  a  positive 
whole  number. 

From  this  we  see  that  the  Binomial  formula  in  its  greatest  gen- 
eralit}^  is  a  special  case  of  Taylor's  formula. 

The  series  (i)  will  be  finite  if  ?i  is  a  positive  whole  number,  but 
not  othenvise. 

When  the  series  (i)  is  infinite  it  should  be  examined  to  see 
whether  it  is  convergent  or  divergent,  for  values  mar  be  given  to 
X,  //,  n,  which  will  render  the  equation  (i)  untrue. 

For  example,  let  x=i,  /^  =  — 3,  and  ;/=  — 2  ;  then  the  left-hand 
member  of  (i)    becomes    (1  —  3)"^  which    equals  (— 2)"^  which 

equals -,  which  equals  \,  a  definite  quantity. 

But  the  right-hand  member  of  (i)  becomes  1+6  +  27+108  +  .  . 
a  sum  of  positive  whole  numbers  each  greater  than  the  one  be- 
fore it,  and  evidently  the  sum  does  not  approach  |. 

20.  Examples. 

1.  Develope  (^i— AJ~^  by  Taylor's  formula. 

2.  Develope  (i—x)~'^  by  the  Binomial  formula. 
Compare  f ij  and  (2')  with  XII,  Art.  8. 


SERIEvS.  195 


J.     Develope  (i-\-xj^  by  Taylor's  formula. 

4.  Develope  (i-\-x)'^  by  the  Binomial  formula. 
Compare  (t^)  and  (^4^  with  XII,  Art.  13,  Ex.  3. 

5.  Develope  (^1  —  2^-/'^  by  the  Binomial  formula. 

<5.     Develope  (a~'2x)   ''  by  the  Binomial  formula, 
-i 

7.  Develope  (c^-^.x'' )   "^   according    to    positiv^e     increasing 

powers  of  x  by  Taylor's  formula. 

8.  Develope  (c^-\-x'')   '^    according    to    positive    increasing 

powers  of  c  by  Taylor's  formula. 

-}- 
g.     Develope  (c^-\-x^)   '^   according    to    positive    increasing 

powers  of  X  by  the  Binomial  formula. 

_i 

10.  Develope   (c^-\-x^)   ^  according    to    positive    increasing 

powers  of  c  by  the  Binomial  formula. 

11.  Find   the   first   negative   term   in   the    development   of 

(i-\-x)  '^  by  the  Binomial  formula. 


CHAPTER  XV. 

LOGARITHMS. 

I.  After  the  extension  of  the  theory  of  indices  in  Chapter  XI  so 
as  to  embrace  incommensurable  exponents,  we  are  enabled  to  give 
an  interpretation  to  the  expression 

a"" 
for  all  possible  values  of  x,  integral  or  fractional,  commensurable 
or  incommensurable.  Since  x  appears  in  this  expression  in  such 
an  unrestricted  form  it  is  common  to  speak  of  the  expression  as  an 
exponential  function  of  x,  intending  to  call  attention  thereby  to  the 
fact  that  X  may  be  considered  a  continuous  variable  as  in  any 
ordinary  algebraic  function. 

If  in  the  equation 

we  assume  x  to  pass  from  one  extreme  of  the  algebraic  scale  to 
the  other,  taking  in  every  possible  value,  then  we  are  able  to  give 
a  meaning  to  this  equation  in  two  variable  ;  because  for  every  pos- 
sible value  of  X,  «',  that  is,  r,  has  a  definite  meaning  and  value. 

In  this  connection  it  must  be  remembered  that  we  are  using  a 
and  a'  under  the  restrictions  mentioned  in  XI,  Art,  15.  So  that 
when  7ve  speak  of  a''  we  mean  that  a  is  a  positive  number,  and  by  the 
value  of  a""  we  mean  that  one  of  its  values  which  is  positive.  Hence 
in  the  equation  «"'=_>'  we  are  to  think  of  but  one  value  of  y  re- 
sulting when  any  particular  value  is  assigned  to  x.  Thus  in 
io*^5=y  ^rg  are  to  understand  i'=  +  ^^  10  and  not  r=  —  >^io  or  any 
other  possible  value  of  y. 

Of  course  the  very  restrictions  just  mentioned  prevent  y  from 
having  a  7iegative  value.  Moreover,  it  is  net  evident  that  y  can 
have  every  positive  value  we  please.  For  example,  is  it  not  plain 
that  a  value  of  x  exists  which  satisfies  the  equation  10*  =  -.  In 
general,  while  it  is  easy  to  see  that  in  the  equation 

there  always  exists  a  value  of  y  for  any  value  assigned  to  x,  it  is 
far  from  evident  that  there  exists  a  value  of  x  corresponding  to 
every  value  which  may  be  assigned  to  r.  Whence  the  necessity 
for  the  following  theorems. 


/ 
Logarithms.  197 

2.  Theorp:m.      77/6'  expression  w  can  be  made  to  differ  from  i 
by  less  than  any  assigned  quayitity  if  x  be  sufficiently  increased. 

Suppose  it  be  required  to  increase  x  so  that 

a^<i-\-d,  (\) 

where  a' stands  for  an  assigned  positive  number.  Then  we  must  have 

(i^dy^a,  (2) 

or,  by  the  binomial  theorem, 

1 .  2 
It  is  easy  to  see  that  i  -\-xd  can  always  be  made  greater  than  a, 
however  small  d  may  be.    Much  more,  then,  will  the  left  member 
of  (^3 J  be  greater  than  a.     In  fact,  the  inequality 

I  +  xd'^  a 
will  hold  if  xd'^a—i, 

or  if  .r>  — y- . 

d 
1 
Hence  to  make  a'  less  than  i+a^take 

«— I 

1 
Example:     Find  .v  such  that   10' <i.oooi. 

Here  ^/=.oooi  and  ^=10  ;  whence 

Q 

.r>  ,  or  Qoooo. 

.0001 

3.  Theorem,    The  expression  a'  is  a  contijinous  foinction  of  x. 
Suppose  a^'-^y  and  let  x  take  on  any  increase,  s,  and  suppose 

the  corresponding  value  oi  y  bej'  +  /,  so  that 

We  are  to  prove  that  as  x  passes  continuously  from  x  to  x-\*s 
that  y  passes  continuously  from  y  to  y-\-t;  that  is,  as  x  changes 
from  X  to   x-\-s  by  passing  over  every  intermediate  value  that  y 
changes  from  r  \.o  y-\-t  by  passing  over  every  intermediate  value. 
The  equation  a'''=y-\-t  may  be  written 

a"a'=y-\-t,  ^        (2) 

and  .since  «'^=j',  this  may  be  written 

a^a'=a^-\-t.  (7,) 

or  a'-'(a'—i)  =  t.  (4.) 


198  Algebra. 

Now,  by  the  last  theorem,  by  taking  .s^  small  enough  a'  may  be 
made  to  differ  from  i  by  an  amount  as  small  as  we  please.  Hence 
in  equation  (^)  t  may  be  made  as  small  as  we  please  by  taking  .9 
small  enough.  That  is,  the  difference  between  two  successive 
values  of  a"-'  can  be  made  as  small  as  we  please.  Therefore  it  is  a 
continuous  function  of  x. 

4.  It  follows  directly  from  the  above  theorem  that  for  every 
positive  value  ivhich  may  be  assigned  to  y  iji  the  equation  a'==y,  a 
correspo7iding  value  of  x  exists  zvhieh  will  satisfy  the  equation. 

For  the  last  article  shows  that  as  x  is  increased  continuoush' 
from  the  value  o  without  limit  that  r  increases  continuously  from 
the  value  i  w^ithout  limit.  That  is,  y  may  have  every  value 
greater  than  i .  It  is  also  seen  that  as  x  is  decreavsed  continuously 
from  the  value  o  without  limit  that  r  decreases  continuously  from 
the  value  i .     That  is,  y  may  have  every  fractional  value. 

The  above  shows  that  if  an}^  value  be  assigned  to  j'  in  the  equa- 
tion a''=y  that  a  value  of  .v  exists  which  will  satisfy  it,  but  it 
does  not  explain  how  to  find  that  value.  Thus  it  does  not  show 
how  to  find  X  in  the  equation  10' =  5.  The  method  of  finding  this 
will  be  explained  later. 

5.  DefinitioNvS.  In  the  equation  a'—y,  where  a  is  some 
chosen  positive  number  not  i  ; 

The  constant  quantity  a  is  called  the  Base. 

The  quantity^  is  called  the  Exporientia!  of  .v  to  the  base  a. 

The  quantity  x  is  called  the  Logarithm  of  y  to  the  base  a,  and 
is  written  x—Xogay- 

The  use  of  the  w^ord  logarithm  may  be  kept  in  mind  by  remem- 
bering this  sentence  :  In  the  equation  a'=^y,  x  is  called  the  Ex- 
ponent of  the  power  of  a  or  the  Logarithm  of  y. 

Of  course  the  two  equations 

a'^y  (i) 

.r=logaj'  (2) 

express  the  same  truth  respecting  the  relation  between  x  and  y. 
The  second  equation  uses  the  logarithmic  notation  and  is  always 
to  be  interpreted  by  means  of  the  first  equation. 

If  in  the  equation  a'=y,  where  a  is  some  positive  number  not 
I,  different  values  be  assigned  to  v  and  the  corresponding  values 


Logarithms.  199 

of  .V  be  computed  and  tabulated,  the  results  constitute  a  System  of 
Logarithms. 

Since  any  positive  value  except  i  may  be  chosen  for  the  base 
a,  the  number  of  different  possible  systems  of  logarithms  is  un- 
limited. In  fact,  however,  only  two  systems  have  ever  been 
tabulated  ;  the  Natural  or  Napcrian  or  Hyperbolic  Syste?n,  whose 
base  is  approximately  2.7182818  +  ,  and  the  Common  or  Briggs' 
System,  whose  base  is  10. 

The  Naperian  logarithms  of  all  numbers  from  i  to  20,000  have 
been  computed  to  27  places  of  decimals.  The  common  logar- 
ithms of  all  numbers  from  i  to  over  200,000  have  been  found. 
They  are  usually  printed  to  seven  decimal  places,  but  they  have 
been  computed  to  many  more. 

The  great  value  of  a  table  of  logarithms  is  the  immense  amount 
of  labor  which  can  be  saved  by  its  use  in  multiplication,  division, 
evolution,  or  involution  of  numbers,  as  will  be  explained  here- 
after. 

5.  Exa:\iplEvS.  Write  the  following  equations,  using  the  log- 
arithmic notation  : 


I.     10*  =  -. 

^• 

io--^=  1.77828-1- 

2.       t''==1'. 

9' 

a""'=!=a"a''. 

J.        11''=  121. 

JO. 

a'  =  a. 

4.        1 0'=  1000. 

II. 

^^^^-''=r. 

5.      1 6 --^=2. 

12. 

J^'^-'^y. 

6.      io'=i. 

13- 

eJ'=a. 

7.      io~^=.ooi. 

^4- 

I  Q  .30:030^  2. 

i^xpress  the  following,  using 

the  exponential  notation  : 

^5-     log^;(i)=-.3333  + 

19- 

logjo24=  10. 

16.     log,  4=. 602060 

20. 

loge  r=  I . 

^7-     log,,ioooo=4. 

21. 

log/,^'^  =  ^. 

18.     logj^.ooooi==  — 5. 

22. 

\ogoa  =  B. 

6.    PkoitErties  of  Logarithms.     Inasmuch   as   logarithms 
are  merely  the  exponents  of  a  fixed  base,  the  properties  of  logar- 


200  Algebra. 

ithms  are  entirely  dependent  upon  the  properties  of  exponents  in 
general,  which  have  already  been  established. 

Among  the  fundamental  properties  of  logarithms  are  these  : 

The  logarithm  in  any  system  of  the  base  itself  is  i . 

For  a^—a, 

that  is,  loga«=i. 

The  logaritlun  of  U7iity  in  all  systems  is  o. 

For  <2"=i, 

that  is,  logai=o. 

Negative  numbers  have  7io  logarithfns. 

For  in  the  equation  a'  =j',  a  is  positive  by  supposition  and  by 
the  value  of  «'  we  mean  that  one  of  its  values  which  is  positive. 
Hence  j^  cannot  be  negative.     See  Art.  i. 

If  we  understand  the  same  S3^stem  of  logarithms  to  be  used 
throughout,  then  the  following  four  theorems  hold. 

7.  Theorem.  The  logarithm  of  the  product  of  several  minibos 
equals  the  sujn  of  the  logay  ithms  of  the  separate  Jaetors. 

Let  n  and  r  be  any  two  positive  numbers  and  let 

log;,7?=A-  and  loga;^=.2'.  fi) 

Then,  by  the  definition  of  a  logarithm, 

71= a'  and    r=a~. 
Multiplying  these  equations  together,  member  by  member,  . 

nr=a'''^~. 
That  is,  \ogan7'=x-^s, 

or,  from  ("i;,  log;,;^r=log,.,;^^-log,,r.  (a) 

In  the  same  way,  if  log;,  5=?/,  then 

7irs=a''"^'''^'". 
That  is,  logn  /zr^= log,,  n  +  log,^  r+  log« .?. 

8.  Theorem.  The  logarithn  of  the  quotie7it  of  two  7iii7nbe7's 
equals  the  loga7'ith77i  of  the  divid-end  minus  the  logarithm  of  the 
divisor. 

Let  71  and  r  be  any  two  positive  numbers  and  let 

log,,  11= X  and  logar=z.  (i) 

Then,  by  definition,        ^^— <2'and    7'=a\ 


Logarithms.  201 


Consequently  -=  — =^' 

r      a^ 


Therefore,  by  definition, 

or,  by  equation  (\), 

log, 


n 
r 


=logfl«— logar.  (b) 


9.  Theorem.  The  logarithm  of  a  powei^  of  a  number  is  equal 
to  the  logarithm  of  the  number  multiplied  by  the  exp07ient  of  the 
power. 

Let  71  be  any  number,  and  let  \o<gan=x.     Then,  by  definition, 

n^a"" . 
Consequently  n^^^a^"- . 

Therefore,  by  definition,     log/j  ?2^=/>-r. 
That  is,  loga;z^=/ loga;2.  (c)  _ 

10.  Theorem.  The  logarithm  of  any  root  of  a  number  is  equal 
to  the  logarithm  of  the  riumber  divided  by  the  index  of  the  root. 

Let  n  be  any  number,  and  let  \Q%an---x.     Then,  by  definition, 

n  =  « ". 

Consequently  's/n—a^. 

Therefore,  by  definition, 

log„(V»)=:^. 
That  is,  log„(V„)='°?«".  (d) 

1 1 .  Theorem.  If  several  numbers  are  in  geometrical  progression!^ 
their  logarithms  are  in  anithmetical progressioii. 

Let  the  numbers  which  are  in  geometrical  progression  be  rep- 
resented by 

71,  7ir,  7ir^,  nr^,  .    .    . 

Then  their  logarithms  to  the  base  a  form  the  series 

loga  n ,  \ogn  n  -f  log,,  r,  log,,  n  +  2  log,,  r,  log,,  w  -}-  3  log;^  r,  .    .    . 
which  is  an  arithmetical  progression  with  the  common  difference 
equal  to  log«r. 

A— 25 


202  Algebra. 

12.  Examples.  In  these  examples  and  in  all  the  following 
pages  the  Common  LogaiitJun  is  designated  by  the  symbol  log 
instead  of  log,^.  Hence  when  no  subscript  appears  we  are  to  un- 
derstand that  the  base  is  lo. 

1.  log  (1888  X476-M492)=log  1888 -flog  476— log  1492. 

2.  log  [V789X_(|ftf)^=ilog  789  +  5  log  239-5  log  930. 

3.  log    i--^^^    =what? 

L      ^  ^^      J 

4.  log/,(<:V^-^//i';/2)=what? 

5.  logzJ(/^^^^^^^)  =  what? 

/•    log^>J":w,^=wbat? 

8.  log/,  ^^= what? 

9.  Prove  loga(loghi5'')  =  loga^. 

10.     Prove  loga^= 


logert' 


13.  Characteristic  and  Mantissa.    For  reasons  which  will 
appear  later  the  common  logarithm  of  a  number  is  always  writ- 
ten so  that  it  shall  consist  of  a  positive  decimal  part  less  than  i 
and  an  integral  part  which  may  be  either  positive  or  negative. 
Thus  the  common    logarithm  of  .0256  is  really  —i. 591 76,  since 


■°""""-ro-W«='o256. 


59176 

But  instead  of  writing 

log.o256=  — 1.59176 
we  write  the  equivalent  equation 

log  .0256=  — 2 -f. 40824, 
or,  as  is  the  universal  custom,  with  the  minus  sign  over  the  2, 
log  .0256=2.40824 
The  minus  sign  over  the  2  shows  that  2  is  alone  affected  ;  that 
is,  the  decimal  fraction  following  it  is  positive.     The  student  must 
always  take  especial  car  e  to  coi'rectly  iyiterpret  this  method  ofyiotation. 
When  the  logarithm  of  a  number  is  arranged  so  that  it  consists 
of  a  positive  decimal  part  less  than  i  and  an  integral  part  either 


Logarithms.  203 

positive  or  negative,  special  names  are  given  to  each  part.  The 
positive  or  negative  integral  part  is  called  the  Chai^aderistic  of  the 
logarithm.     The  positive  decimal  part  is  called  the  Maiitissa. 

14.  The  following  table  is  self-explanatory: 

ic*  =10000,   whence   log  10000=4 
10''  =1000,  "         log  1000=3 

10-  =100,  "         log  100=2 

10'  =10,  "         log  10=1 

10"  =1,  "         log  1=0 

io~'=.i,  "         log  .1  =  — I 

io~-=.oi,  "         log  .01  =  — 2 

io~^=.ooT,  "         log  .001  =  — 3 

io~'*=.oooi,  "  log  .0001  =  — 4 
Here  we  observe  that  as  the  numbers  pass  through  the  series 
loooo,  1000,  100,  10,  etc.,  the  logarithms  pass  through  the  series 
4,  3,  2,  I,  etc.;  that  is,  continuous  division  of  the  number  by  10 
corresponds  to  a  continuous  subtraction  of  i  from  its  logarithm. 
This  can  easily  be  shown  to  hold  in  any  case. 

15.  Theorem.  Multiplyirig  any  ymmbcr  by  10  increases  the  com- 
mon logarithm  by-  i,  and  dividing  any  number  by  10  decreases  its 
common  logai-ithm  by  i. 

Let  r  be  any  number  and  x  its  common  logarithm.     Then 

logj'=jf, 
or  io^=j'. 

We  are  to  prove  log  iov=.r-fi, 

and  log  T  0 J' = -V  —  I . 

By  formula  (a).  Art.  7, 

log  ioj'=log  j'.+  log  10. 
But  log  10=1  (Art.  6)  and  log  i'=a-. 

Hence,  substituting,        log  ioj'=-t-j-i. 
Also  by  formula  (b).  Art.  8, 

That  is,  log  iVj'=-^  —  T- 


204  Algkbra. 

16.  Corollary.  Moving  the  decimal  point  iyi  a  numbei'  one 
place  to  the  right  increases  its  common  logarithm  by  i,  and  moving 
it  one  place  to  the  left  decreases  its  logarithm  by  i. 

17.  Corollary.  The  common  logarithms  of  all  7iumbers  con: 
sisting  of  the  same  significant  fgnres  have  the  same.  ma?itissa. 

For  moving  the  decimal  point  merely  adds  or  subtracts  i  from 
the  logarithm  ;  that  is,  merely  affects  the  characteristic.  Thus 
log  256=2.40824 
log  25.6=1.40824 
log  2.56=0.40824 
log  .256=1.40824 
log  .0256=2.40824 
log  .00256=3.40824 

18.  Theorem.  If  a  member  has  its  frst  significant  figure  in 
units'  place,  the  characteristic  of  its  common  logarithtn  is  o. 

If  the  number  has  its  first  significant  figure  in  units'  place,  the 
value  of  the  number  must  lie  somewhere  between  i  and  10.  But 
the  logarithm  of  i  is  o  and  the  logarithm  of  10  is  i.  Hence  the 
logarithm  of  the  proposed  number  must  lie  somewhere  between  o 
and  I.   (Art.  3.)     That  is,  its  characteristic  must  be  o. 

Thus  log  2.56=0.4082400 

and  log  9-99=o-9995655 

19.  Theorem.  The  characteristic  of  the  conwiori  logarithm  of  a 
number  eguals  the  mimber  of  places  the  first  significant  figiire  of  the 
number  is  removed  from  units'  place,  and  is  positive  if  the  first  sig- 
nificayit  figure  sta?ids  to  the  left  of  units'  place  and  is  negative  if  it 
stands  to  the  right  of  units'  place. 

By  the  previous  article,  if  the  first  significant  figure  stands  in 
units  place  the  characteristic,  is  o.  If  the  first  significant  figure 
stands  in  the  ni\i  place  to  the  left  of  units  place,  then  the  char- 
acteristic of  its  logarithm  must  be  a  number  such  that  it  can  be  made 
from  o  by  adding  i  to  it  n  times.  (Art.  15.)  In  other  words,  the 
characteristic  must  be  n. 

If  the  first  significant  figure  of  the  given  number  stands  in  the 
;^th  place  to  the  right  of  units  place,  then  the  characteristic  of  its 


Logarithms.  205 

logarithm  must  be  a  number  such  that  it  can  be  made  by  sub- 
tracting I  from  o  n  times  ;  that  is,  it  must  be  —n. 

20.  Examples.  The  above  enables  us  to  tell  by  inspection 
the  characteristic  of  the  common  logarithm  of  any  number.  Thus 
the  characteristic  of  the  logarithm  of  237945.834  is  +5,  because 
2,  the  first  significant  figure,  stands  in  the  fifth  place  to  the  left 
of  units  place.  In  the  same  way,  the  characteristic  of  the  log- 
arithm of  .0007423  is  —4,  because  7  stands  in  the  fourth  place  to 
the  right  of  units  place. 

In  determining  the  characteristic,  care  must  be  taken  that  we 
count  from  the  units  place  and  not  from,  the  decimal  point ;  for  the 
decimal  point  stands  to  the  right  side  of  units  place. 

Find  the  characteristic  of  the  logarithms  of  the  following  num- 
bers : 

T.      1888. 1 19  5.     3000.0303 

^.     .3724  6.     .00000000849 

J.     783294.009  7.     .00010000849 

</.      .0084297  •  8.     3.00007 

21.  Tables  of  Common  Logarithms.  The  nature  of  the 
work  in  which  logarithms  are  to  be  used  determines  the  size  and 
accuracy  of  the  tables  which  should  be  employed.  For  some  pur- 
poses a  table  of  the  logarithms  of  all  numbers  from  i  to  loooo, 
given  to  five  or  six  decimal  places,  is  sufficient.  For  many  pur- 
poses a  table  of  the  logarithms  of  all  numbers  from  i  to  1 00000 
to  seven  decimal  places  is  desirable,  and  this  may  be  said  to  be 
the  standard  table.  We  print  herewith  a  sample  page  from  such  a 
table.  This  page  contains  the  logarithms  of  all  numbers  between 
25600  and  26100,  or,  more  correctly,  the  mantissas  of  the  log- 
arithms of  these  numbers.  For,  since  the  characteristic  of  the 
common  logarithm  of  anj'  number  can  alwaj^s  be  found  by  inspec- 
tion, characteristics  are  never  printed  in  such  tables. 

Suppose  we  wish  to  find  the  logarithm  of  25964.  Then  know- 
ing that  the  characteristic  of  the  logarithm  of  25964  is  4,  we  will 
fiud  the  mantissa  from  the  table.  We  run  down  the  column  headed 
Num.  until  we  come  to  the  figures  2596.    We  then  run  across  the 


2o6 


Algebra. 


A  sample  page  from  a  table  of  logarithms: 


Num. 

(il 
G2 
63 
64 

(>5 
6G 
(>7 
68 
69 

25  70 
71 
72 
78 

74 

75 
76 

77 
78 
79 

25  80 
81 
82 
88 
84 

85 
86 
87 
88 
89 

25  90 
91 
92 
93 
94 

95 
96 
97 
98 
99 

26  00 
01 
02 
03 

04 

05 
06 
07 
08 
09 

Num. 


0 


3 


408  2400:2569 
40964265 
5791  5961 

7486  7656 
t  9180  9350| 

409  0874  1043 
2567  2736' 
42594428 
5950:6119 
7641:7810 


2739 
4435 
6130 
7825 
9519 

1212 
2905 
4  597 

6288 
7979 


t  9331  950ul9669 

410  1021  1190  1359 
2710  28788047 
4808  45(57  4785 
6085  6254 1 6 423 

7772i7941i8nO 
t  9159  9627  979() 

411  1144  1313  1481 
2829  :^998  8166 
4518  4682  4850 

619716865  6534 

7880  8048  8217 

t  9562  97819899 

412  124411412  I58u 
29i;5  8098|3261 

46O5I4773U94I 

6285 '6458  6621 

7961  8182  88U0 

+  9648i9811  9978 

413  1321 '1488  1656 


2909  3078  3248  3417  3587 
4604  4774  4944  51 13 '5283 
6300|6469  6639  6808  6978 
7994  8 164  18338  8503  8672 
9688  985cS* 027 1*196  * 366 


3757 
5452 
7147 

8841 
*535 


J382  155111720^1889  2059:2228 
8074  3243  8413  3582  3751  8920 
4766  4985  5105j5274  5448  5612 
6458  6627  6796  6965  7184 i7808 
8148  8317  8486  8655  8824  8998 


9838  *007 


176*845*5141*683 


1527  1696  18(35:2034  2203 
3216  3885  8554  3723  8891 
4904  5078  5242l54i0  5579 
6592  6760  6929  7098  7266 


8278  8447  8616 
9964  *  188  *3()1 
1»)50  1818  1987 
8384  85Uh!8671 
5019 15187  5355 


6702|6870 
8885' 8558 
*()67  *285 
1748^1917 
8429  8597 


5109:5277  5445 
6789  «)957  7125 


7039 
8721 
*  403 
2085 
8765 


8468  8636 
*14U  *;U4 
1824  1991 


299813165  3333  3501  3668 
4674!4842i5(M)9  5177  5845 
6850  ti5  18  668516858  7U20 
8025  8193!8360!8528  «695 
9700:9867  *085  *202  *369 


414  1374 
3047 
4719 
6891 
806ci 

f  9733 

415  1404 
3073 
4742 
6410 

8077 
t  9744 

416  1410 
3076 
4741 


1541  1708 
821 41338 1 
4887  5054 
655916726 
8230;  8897 


1876 
8549 
5221 
6898 
8564 


9901  *  068  *235 
1570  1737|1904 
3240  34<»7:8574 
4909  5075 '5242 
6577  6743 '69 10 


8244 
9911 
1577 
3242 
49u7 


841118577 
*077|*244; 
1743 '19  10 
8409  3575 
5074  5240 


2043 
871 6 
5388 
/060 
8731 

*402 
2071 
3741 
54U9 
7077 

8744 
*4ll 
2077 
8742 
5407 


8804 
*482 
2159 


8784 

*4ro 

2155 
3840 
5524 

7207 
8890 
*571 
2253 
8938 

5613 
7298 
8971 
*649 
2327 


8958 
*639 
2;:l24 
4008 
5692 

7375 
9058 
*740 
2421 
4101 

5781 
7461 
9139 
*817 
2495 


3836  4004  4171 
5512  5680  5847 


7188 
8868 
*537 

2210 
8883 
5556 
7227 
889S 


7855  7528 


2872 
4060 
5748 
7485 

9121 

*807 
2492 
4177 
5860 

7544 
9226 
*9()8 
2589 
1269 

5949 
7629 
9807 
*985 
2662 


3926 
5622 
7817 
9011 
*704 

2397 
4089 
5781 
7472 
9102 

*852 
2541 
4229 
5917 
7604 

9290 
*976 
2661 
4  34  5 
6029 

7712 
9894 
1076 
2757 
4487 

6117 
7796 
9475 
1153 
2830 


Diff.  j  Dilf,  A:  Mul. 


4339  4507 
6015  6182 

7690|7858 


9030  9197  9865  9582 
*704*872'l039'l206 


2378  2545 
4051  42l« 
5723  5890 
7894i7561 
9065  9282 


*  569  *  786  *903 
2 238  2405  2572 
3907  4074  4  241 
557615743  5909 
7244i7410  7577 
I     I     1 


8911 
*577 
2248 
8908 


9077  9244 
*744*911 
2410  2576 
40754241 


2712'2S80 
4385  4552 
6057  6224 
77>9  7896 
9899  9566 

1070  1237 
2739  2906 
4408  4575 
607(5  (3248 
7744  7911 

9411  9577 
1077  1244 
2743  2909 
440.S  4574 
6072  6239 


167 


1 

169 

2 

338 

3 

507 

4 

676 

5 

845 

() 

1014 

7 

1183 

8 

1352 

9 

1521 

1 

168 

2 

33(> 

3 

504 

4 

672 

0 

840 

6 

1008 

7 

1176 

8 

1844 

9 

1512 

1 

167 

2 

384 

3 

501 

4 

668 

5 

835 

(> 

1(J02 

7 

1169 

8 

1836 

9 

1  ^^^y.^ 

0 


3 


9   Diff.  Diff.  &  Mul. 


Logarithms.  207 

page  on  the  same  horizontal  line  with  2596  until  we  come  to  the 
column  headed  4,  at  which  place  will  be  found  the  figures  3716, 
which  are  the  last  four  figures  of  the  required  mantissa.  The  first 
three  figures  are  found  in  the  column  headed  o  and  are  seen  to  be 
414.  Whence  the  mantissa  of  the  logarithm  of  25964  is  .4143716, 
and  therefore 

log  25964=4.4143716. 

Of  course  a  decimal  point  belongs  before  the  mantissa  of  each 
logarithm,  and  since  this  fact  is  understood,  it  is  unnecessary  to 
print  the  decimal  points  in  a  table. 

Inasmuch  as  the  first  three  figures  of  the  mantissa  are  only 
printed  in  the  column  headed  o,  it  is  necessary  to  mark  the  point 
at  which  these  first  three  figures  change.  This  is  done  by  an 
asterisk  ( *)  standing  in  the  place  of  a  cipher  in  the  last  four  fig- 
ures. Thus  to  find  the  logarithm  of  25646  we  must  note  that  the 
three  figures  change  from  408  to  409  at  the  point  25645  (which  is 
indicated  by  printing  ^027  in  place  of  0027),  and  consequently 
log  25646=4.4090196. 

The  dagger  (t)  which  appears  in  column  o  is  intended  to  cau- 
tion us  that  the  first  three  figures  change  at  some  place  in  the 
same  horizontal  line  with  it. 

If  we  wish  to  find  the  logarithm  of  a  number  consisting  of  more 
than  five  figures,  say  25705.84,  then  we  must  take  the  nearest 
number  whose  logarithm  is  given  in  the  table,  that  is  to  say, 
25706.00.     Thus 

log  25705.84=4.4100345,  nearly. 

Greater  accuracy  may  be  secured  by  means  of  tables  of  dif- 
ferences and  multiples,  as  is  explained  in  connection  with  any 
good  table  of  logarithms. 

A  table  of  logarithms  of  numbers  from  i  to  1 00000  can  be  used 
to  find  the  logarithm  of  any  number  consisting  of  five  significant 
figures.  Thus  to  find  the  logarithm  of  25.964  we  entirely  neglect 
the  decimal  point-  in  finding  the  mantissa,  as  the  decimal  point 
affects  the  characteristic  alone.     Thus 

log  25.964=1.4143716. 

A  table  of  logarithms  also  enables  us  to  find  the  number  corre- 
sponding to  any  given  logarithm  by  a  mere  reversal  of  the  process 
already  explained.     Thus  all  numbers  the  mantissas  of  whose 


2o8  Algebra. 

logarithms  lie  between  .4082400  and  .4166239  are  on  the  speci- 
men page  we  give.  Suppose,  as  an  example,  that  w^e  wish  to 
find  the  number  corresponding  to  the  logarithm  2.4127469.  The 
characteristic  merely  affects  the  decimal  point,  and  consequently 
the  problem  is  merely  to  find  the  significant  figures  which  cor- 
respond to  the  given  mantissa.  The  nearest  mantissa  printed  in 
the  table  is  .4127461  and  this  corresponds  to  the  figures  25867. 
Hence,  pointing  oif  the  number  by  means  of  the  characteristic,  we 
find  that  the  number  whose  logarithm  is  2.4127469  is  258.67. 
In  connection  with  tables  of  logarithms  methods  are  explained  by 
means  of  which  more  figures  of  this  number  could  be  found  by 
means  of  tables  of  multiples  and  differences,  or  of  proportional  parts. 

22.  KXAMPLKS. 

1.  Find  the  logarithm  of  25734. 

2.  Find  the  logarithm  of  26000000. 
J.     Find  the  logarithm  of  25.999 

/.  Find  the  logarithm  of  .02578411 

5.  Find  the  logarithm  of  .260099 

6.  Find  the  number  whose  logarithm  is  3.4147561 

7.  Find  the  number  whose  logarithm  is  0.4104400 

8.  Find  the  number  whose  logarithm  is  2.415999 
g.  Find  the  number  Avhose  logarithm  is  1.4094094 

10.     Find  the  number  whose  logarithm  is  7.4100000 

23.  Multiplication  by  Logarithms.  Formula  {a)  (Art.  7) 
enables  us  to  find  the  product  of  several  numbers  by  means  of  a 
table  of  logarithms.  Thus,  suppose  we  wish  the  product  of  98 
by  265.     From  a  table  of  logarithms  we  find 

log  98=  1.9912261 

log  265=  2.4232459 

log  98  X  265=4.4144720 

From  the  table  of  logarithms  (see  sample  page)  it  is  found  that 

4.4144720  is  the  logarithm  of  25970.     Therefore  98  X  265  =  25970. 


Logarithms.  209 

24.  ExAMPi^ES  IN  Multiplication. 

/.  Log  327.45=2.5151450  and  log  79.493=1.9003839;  find 
the  product  of  327.45  x  79.493. 

2.  Log  .53927=1.7318063  and  log  4.7655=0.6781085:  find 
the  product  of  .53927  X  4.7655. 

J.  Log  6.3274=0.8012253  and  log  1645.6=3.2163243  ;  find 
the  product  of  6.3274  x  1645.6. 

25.  Examples  in  Division.     See  Art.  8. 

1.  Find  the  quotient  of  327.45  by  1645.6. 
From  a  table  of  logarithms  we  find 

iQg  327.45=  2.5151450 

log  1645.6=  3.2163243 

log  327.45^1645.6=1.2988207 
It  is  seen  from  a  table  of  logarithms  that  the  number  corre- 
sponding  to   the   logarithm    1.2988207    is    .19901+     Therefore 
327:45 -T- 1645.6= .  19901  + 

2.  Log  53.927=1.7318063  and  log  2.0724=0.3164736;  find 
the  quotient  of  53.927-^2.0724. 

3.  Log  33333=4.5228744  and  log  13001  =  4.1139768;  find 
the  value  of  fetf 

^.  Log  54321  =  4.7349678  and  log  20.877  =  1.3196681  ;  find 
the  value  of  54321^20.877. 

26.  Examples  in  Involution.   See  Art.  9. 

1.  Find  the  third  power  of  1373.3. 
From  a  table  of  logarithms  we  find  that 

log  i373-3=3-t377654 

3 

whence  log  (1373.3)^=9.4132962 

From  the  sample  page  it  is  seen  that  the  number  whose  logar- 
ithm is  9.4132962  equals  2590000000,  nearly.     Therefore 
■^  (1373-3)^=2590000000,  nearly. 

2.  Find  the  fifth  power  of  1.9201,  whose  logarithm  is 
0.2833238. 

A-2(J' 


2IO  Algebra. 

J.  Find  the  tenth  power  of  .69353,  whose  logarithm  is 
1. 8410653. 

/.  Find  the  seventh  power  of  15.926,  whose  logarithm  is 
1. 2021067. 

27.  ExAMPLKS  IN  Evolution.    See  Art.  10. 
/.     Find  the  cube  root  of  26. 

From  a  table  of  logarithms  we  find 

log  2^=1.4141374 
Therefore  log  x^^26=o.47i379i 

The  number  whose  logarithm  is  0.471 3791  is  found  to  be 
2.9606  + 

Hence  ^26=2.9606-!- 

2.  Find  the  square  root  of  668.63,  whose  logarithm  is 
2.8251859. 

J.  Find  the  fifth  root  of  11 09600000000,  whose  logarithm 
is  12.0451664. 

^.  Find  the  tenth  root  of  1.384,  whose  logarithm  is 
0.1411675. 

28.  Exponential  Series.  The  Exponential  Series,  or  the 
Exponential  Theorem,  as  it  is  often  called,  is  an  expansion  of  ^' 
in  terms  of  the  ascending  powders  of  x.  The  following  demonstra- 
tion* of  this  important  theorem  is  due  to  Mr.  J.  M.  Schaeberle,  of 
the  Lick  Observatory,  and  is  inserted  here  with  his  pennission. 

We  are  required  to  expand  a""  in  a  series  of  ascending  powers  of 
X.     Assume 

«^  =  A  +  B.r+C-r^  +  D-x-3  +  E-:i-'+  ...  (i) 

where  A,  B,  C,  etc.,  are  undetermined  coefiicients. 

The  limit  of  the  left-hand  side  of  this  equation  as  x  approaches 
o  is  plainly  i.     The  limit  of  the  right-hand  side  of  this  equation 
as  X  approaches  o  is  A.      (See  XI,  Art.  26.) 
Therefore  A=i. 

Substituting  this  value  of  A  in  (\)  and  then  squaring  both 
members, 
ij        ^^^-*•=I-f2B.^--f(2C-|-B0-^"+(2D-f  2CB).r3 

4-(2E  +  2DB  +  0'»+  .    .    .      (2) 

*See  Annuls  of  Mathematics,  Vol.  Ill,  p.  15J. 


< 


Logarithms.  211 

But  if  we  substitute  2X  in  place  of  x  in  equation  (i)  we  obtain 

Therefore,  equating  like  powers  of  jr  in  equations  (2)  and  (2,)y 
we  obtain 

B=  B^  B^ 

B=B;     C=      ;     D=  -      ;     E=        ;  etc. 

Whence,  on  substituting  these  values  of  B,  C,  D,  etc.,  equation 
(\)  becomes 

|_2         1 3         1 4 
Now,  there  must  exist  some  quantity,  c,  at  present  unknown 
in  value,  such  that 

c^'=^^  (5) 

or,  m  other  words,  such  that 

loge«=B.  (6) 

Substituting  loge^a-  for  B  throughout  equation  (/[)  we  obtain 

a^=i4-^log..  +  ":^^^^^^%-:^^^^+^^^^^    .  .   (7) 

\3  [3  |4 

which  is  called  the  Expo7icntial  Theorem  or  Series. 

29.  To  FIND  THE  Value  of  the  Base  e.  The  base  a  in  the 
last  article  is  any  chosen  positive  quantity  not  i,  and  its  value  is 
therefore  at  our  disposal.  Hence  in  the  exponential  series  (equa- 
tion 7)  we  may  put  a==e,  $0  that  loge^  becomes  loge^  ;  that  is,  i. 
Equation  (-] )  then  becomes 

X^  X^  X* 

^='+-+L.-+i:3  +  |4  +  ---        ^'^ 

This  important  result  is  convergent  for  all  values  of  .v,  (see 
XIV,  Art.  12,  Ex.  13,)  and  consequently  the  equation  is  true 
when  x=  i .     Therefore  we  have 

.=  :  +  :+ |>|i  +  ,;+...  (.) 

By  taking  a  sufficient  number  of  terms  of  this  series  we  may 
approximate  the  value  of  e  to  any  desired  degree  of  accuracy. 
Thirteen  terms  of  the  series  give  ten  places  of  decimals  correctly 
and  we  h^ve 

^=2.7182818284  ...  (2,) 


212  AI.GEBRA. 

This  number  is  one  of  the  most  important  constants  in  mathe- 
matics. It  is  called  the  Naperia7i  Base  and  is  always  represented 
by  the  letter  e.  Its  value  is  known  to  more  than  260  decimal 
places. 

30.  Logarithmic  Series.  The  Logarithmic  Series  is  the  ex- 
pansion of  logeri+-^^  in  terms  of  the  ascending  powers  oi  x. 

From  the  exponential  series 

.-'==i  +  rlog..+-^^^^7^^^-V^^^^^  .    .(.)    . 

Whence,  transposing  the  i  and  dividing  through  byj', 

-^-^-=log..-fr-i^^^V-^-^^^^^^^^      ...   I      r^' 

y  (     h  1 3  S 

Therefore,  since  these  variables  are  always  equal,  , 

limit  (a-'-i,  _limit  \  ,     r(log£fl:+-y(l°S?fI'+        '\\  / .,) 

Whence  it  is  easy  to  see  ' 

limit  f^'  — I)      ,  ,    . 

Now  put  I  -f-^t:  for  <2,  then  we  have 

.        ,,      ,     limit  {(i-^xy-i-) 

logeri+x;-,, :  oj — -y — j- 

Expanding  (\-\-xy'  by  binomial  formula, 
.J  '  o(  1 .  2  1.2.3 

}  rs; 

The   limit  of  the  right  member  as   r  ]:  o  can  be  plainly  seen  ; 
whence  we  obtain  the  equation 

iog.rn-x;=x--V---^+ ...  r6; 

234 

This  is  the  Logarithmic  Se7'ies. 

31.  CoNVERGENCY  OF  THE  SERIES.  The  above  seri,f;s  is  not 
convergent  for  values  of  x  greater  than  i ,  and  hence  cannot  be 
used  for  computing  the  logarithm  of  any  integral  number  but  2. 
The  following  scheme  will  give  a  series  which  is  available  for 
computing  the  logarithms  of  all  integers. 


1.2.3.4 


IvOGARITHMS.  213 

32,  A  Logarithmic  Series  Convergent  for  Integral. 
\'ai.uks  of  .v.     In  the  logarithmic  series 

234 

Substitute  —x  for  x  and  we  shall  have 

X^       X^       JT* 
\oge(i—^')=—JC —      —..     .  (2) 

Subtracting  (2)  from  (i ),  observing  that  \oge( ^ -\-x)—\oge(i—^) 
=  loo:e  ,  we  obtain 

I  —X 

loge.'-"^-^"=2U-  +  V+'-^^H-'A'^-f-    ...     I  (Z) 

i--^'        13  5         7  J 

I  2Z-\-2  2Z 

Now  put  A  =  — -— ,    whence    i4--i'= — ; — ,    i— -r=  ,  and 

2,3 -f  I  23- -f- I  22-\-\ 

"  =        ".     Therefore  we  obtain 

I— .V  Z- 

Whence,  since  loge     /^=loge(^i-f^y)— loge^-,  by  substituting  and 
transposing  logeS"  we  have 

This  series  converges  rapidly  for  integral  values  of  z.  Its  use 
in  computing  the  logarithms  of  numbers  will  now  be  explained. 

33.  To  Coinipute  the  Naperian  Logarithms  of  Numbers. 
The  logarithm  of  i  is  o  in  all  S3'stems.  To  compute  loge2,  put 
3=1  in  equation  (^)  above.     We  then  obtain 


Now  put  2  =  2  in  equation  (^).     Then  we  have 
loge3=. 6931472  +  2 


"  +  --S+    '+-'--+    '-^+.    =1.0986123 
5      3-5'     5-5^     7-5'     9-5'      J 


To  find  log(>4  we  know  loge4=loge2^=2  loge2  ;  whence 
loge4  =  i-3862944 


"H ,4-       ,+-'"-+  .  .  .  !  =  1.6094379. 

9      3-9'     5-9-^     7-9'  J 


214  Algebra. 

To  find  logt>5,  put  ^=4  in  equation  (5).     We  then  have 

loge5=i-3862944+2 

In  like  manner  the  logarithms  of  all  numbers  may  be  found. 
The  logarithms  of  composite  numbers  need  not  be  computed  by 
the  series,  since  the  logarithm  of  any  composite  number  can  be 
found  by  adding  the  logarithms  of  its  component  factors. 

34.  REiyATioN  Between  the  IvOGarithms  of  the  Same 
Number  in  Different  Systems. 

Consider  the  systems  whose  bases  are  a  and  e.  Then  if  71  is  any 
number,  we  wish  to  find  the  relation  between  logon  and  \og,.,n. 

Let  x=loge??  and  r=log;,;z. 

Then  71= e""  and  71  =  a''; 

whence  ^'  =  «-'.  (i) 

Therefore  a^e-''.  (2) 

If  we  write  this  in  logarithmic  notation  we  have 

loge«=-,  (2,} 

or,  substituting  the  values  of  x  and  y,  we  obtain 

Therefore  \ogan=-  ■    -\ogen,  (^) 

loge« 

which  is  the  relation  between  log..,w  and  \ogen. 

35.  Modulus  of  Common  Logarithms.  If  in  equation  (^)' 
above  we  understand  e  to  represent  the  Naperian  base  and  a  the 
common  base,  then  equation  ( ^)  becomes 

^^^^^=iog;io^^^^^^-  ^'^ 

But  Iogeio=loge2  +  loge5  =  (by  Art.  33)    2.3025851  and  - 

log  e  I O 

=  .43429448     Therefore  representing  .43429448  by  M  we  have 
log  71=-1A  \oge7i.  (2) 

The  decimal  represented  by  M  is  known  to  282  decimal  places. 
and  is  called  the  Modulus  of  the  system  of  common  logarithms. 


Logarithms.  215 

Equation  (^2J  is  seen  to^  express  the  important  truth  that  the 
common  logarithm  of  any  niunber  can  be  obtained  by  multiplyini^- 
the  Naperian  logarithin  of  that  number  by  the  modulus  of  the  com- 
mo7i  system. 

36.  Computation  of  Common  Logarithms.  We  can  now 
compute  the  common  logarithms  of  numbers.  We  merely  need  to 
multipl}^  each  of  the  Naperian  logarithms  already  found  by  the 
modulus  ,43429448  .  .   In  this  manner  we  find 

log  2=0.3010300 
log  3  =  0.4771213 
log  4=0.6020600 
log  5  =  0.6989700 
etc.  etc. 

How  can  you  find- log  6  ? 

37,  Historic aIj  Note.  The  invention  of  logaritliins  is  regarded  as  one 
of  the  greatest  discoveries  in  mathematical  science.  The  honor  of  the  inven- 
tion as  well  as  of  the  construction  of  the  .^rst  logarithmic  table  belongs  to  a 
Scotchman,  John  Napier  (1550-1617),  baron  of  Merchiston.  His  first  work, 
Mirifici  logarifhmorum  canonis  descriptio,  appeared  in  1(514  and  contained  an 
account  of  the  nature  of  logarithms  (from  his  standpoint)  and  a  table  of  natural 
sines  and  their  logarithms  to  seven  or  eight  figures.\But  Napier's  logarithms 
were  not  the  same  as  those  now  called  Naperian  logarithms.  The  base  of  his 
system  was  not  e,  although  closely  related  to  it. 

Henry  Briggs,  professor  of  geometry  at  Gresham  College,  London,  was  much 
interested  in  Napier's  invention  and  in  lO  15  visited  Napier  and  suggested  to  him 
the  advantages  of  a  system  of  logarithms  in  which  the  logarithm  of  1  should 
be  0  and  the  logarithm  of  10  should  be  1.  Napier,  having  already  thought  of 
the  change,  gave  Briggs  every  encouragement  to  compute  a  system  of  the 
new  logarithms  and  made  many  important  suggestions,  among  which  was  that 
of  keeping  the  mantissas  of  all  logarithms  positive  by  using  negative  char- 
acteristics. In  1()17  Briggs  published  the  common  logaiithms  of  the  first  1000 
numbers,  the  book  being  QaWtid.  Logariihmornm  chilias  priim.  Briggs  con- 
tinued to  labor  at  the  calculation  of  logarithms,  and  in  1624  published  his 
ArHhmetica  Logarithmiea,  which  conlaimMl  the  logarithms  of  the  numbers 
from  1  to  20000  and  from  90000  to  100000  to  14  places  of  decimals.  This  gap 
between  20000  and  90000  was  filled  up  by  Adrian  Vlacq,  who  published  in  162h 
the  logarithms  of  the  numbers  from  1  to  100000  to  ten  places.  Vlacq's  table 
is  the  source  from  which  nearly  all  the  tables  have  been  derived  which  havo 
^subsequently  been  published. 


2l6  Al^GEBRA. 

The  moaning  of  logarithms  to  Napier  and  Briggswas  enth-ely  differoiit  from 
that  we  now  have.  They  never  thouglit  of  connecting  logarithms  with  the 
idea  of  an  exponent,  and  consequently  had  no  conception  of  what  we  call  the 
base  of  the  system.  Their  idea  of  logarithm  is  contained  in  the  meaning  of 
the  term  itself,  which  comes  from  two  Greek  words  meaning  the  number  of 
the  ratios.  This  idea  of  a  logarithm  is  thus  explained:  Suppose  the  ratio  of 
1  to  10  be  divided  into  a  large  number  of  equal  ratios  (or  factors),  say  1000000. 
Then  it  is  true  that  the  ratio  of  1  to  2  is  composed  of  301030  of  these  equal  ratios 
(or  factors),  and  301030,  the  number  of  the  ratios,  is  the  logarithm  of  2.  In 
the  same  way  the  ratio  of  1  to  3  is  composed  of  477121  of  these  equal  ratios 
(or  factors),  and  the  logarithm  of  3  is  hence  said  to  be  477121. 

The  first  methods  used  for  computing  logarithms  were  very  tedious .  The 
great  work  of  computing  was  finished  long  before  the  discovery  of  the  log- 
arithmic series. 

The  above  note  is  derived  from  J.  W.  L.  Glaisher's  article  on  Logarithms 
in  the  Encyclopedia  Britannica. 


THE   END. 


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